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Add Group_sum_elim.

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favonia 2017-06-07 12:30:08 -06:00
parent 9f1df6becb
commit 1fee1395ed
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7
algebra/product_group.hlean
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@ -67,6 +67,13 @@ namespace group
definition product_inr (G H : Group) : H →g G ×g H :=
homomorphism.mk (λx, (one, x)) (λx y, prod_eq !one_mul⁻¹ !refl)
definition Group_sum_elim (G H : Group) (I : AbGroup) (φ : G →g I) (ψ : H →g I) : G ×g H →g I :=
homomorphism.mk (λx, φ x.1 * ψ x.2) (λx y, calc
φ (x.1 * y.1) * ψ (x.2 * y.2) = (φ x.1 * φ y.1) * (ψ x.2 * ψ y.2)
: by exact ap011 mul (to_respect_mul φ x.1 y.1) (to_respect_mul ψ x.2 y.2)
... = (φ x.1 * ψ x.2) * (φ y.1 * ψ y.2)
: by exact interchange I (φ x.1) (φ y.1) (ψ x.2) (ψ y.2))
definition product_functor [constructor] {G G' H H' : Group} (φ : G →g H) (ψ : G' →g H') :
G ×g G' →g H ×g H' :=
homomorphism.mk (λx, (φ x.1, ψ x.2)) (λx y, prod_eq !to_respect_mul !to_respect_mul)