Theorems · Theorem · functional analysis
RCLike.is_real_TFAE
∀ {K : Type u_1} [inst : RCLike K] (z : K),
[(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(RCLike.re z) = z, RCLike.im z = 0, IsSelfAdjoint z].TFAEThere are several equivalent ways to say that a number z is in fact a real number.
- Defined in
- Mathlib.Analysis.RCLike.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RCLike
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Realstatement and proof · cited by 25,697
- RingHomstatement · cited by 10,189
- AddMonoidHomstatement · cited by 3,230
- RCLikestatement and proof · cited by 2,829
- add_zeroproof · cited by 2,707
- MulZeroClass.mul_zeroproof · cited by 2,091
- MulZeroClass.zero_mulproof · cited by 1,625
- sub_selfproof · cited by 996
- starRingEndstatement and proof · cited by 671
- IsSelfAdjointstatement and proof · cited by 545
- RCLike.ofRealstatement and proof · cited by 350
Cited by5
Results whose statement or proof uses this declaration.
- RCLike.conj_eq_iff_reproof · cited by 5
- RCLike.conj_eq_iff_improof · cited by 4
- inner_self_ofReal_reproof · cited by 3
- RCLike.im_eq_zero_iff_isSelfAdjointproof · cited by 1
- RCLike.conj_eq_iff_realproof · cited by 1