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Theorems · Theorem · group theory

SemidirectProduct.map_left

∀ {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [inst : Group N₁] [inst_1 : Group G₁]
  [inst_2 : Group N₂] [inst_3 : Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂)
  (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn)
  (g : N₁ ⋊[φ₁] G₁), ((SemidirectProduct.map fn fg h) g).left = fn g.left
Defined in
Mathlib.GroupTheory.SemidirectProduct
Cited by
0 results in Mathlib
Foundations
Depth 37 from the axioms · uses propext, Quot.sound
Assumes
GroupGroupGroupGroup
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DFunLike.coe · cited by 62936DFunLike.coeGroup · cited by 6238GroupMonoidHom · cited by 3629MonoidHomMonoidHom.comp · cited by 469MonoidHom.compMulAut · cited by 158MulAutMulEquiv.toMonoidHom · cited by 126MulEquiv.toMonoidHomSemidirectProduct · cited by 69SemidirectProductSemidirectProduct.left · cited by 30SemidirectProduct.leftSemidirectProduct.map · cited by 8SemidirectProduct.mapSemidirectProduct.map_leftCITED BYCITES

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