isghm3 - Metamath Proof Explorer

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Theorem isghm3 19257

Description: Property of a group homomorphism, similar to ismhm 18820. (Contributed by Mario Carneiro, 7-Mar-2015.)

**Hypotheses**
Ref	Expression
isghm.w	⊢ 𝑋 = (Base‘𝑆)
isghm.x	⊢ 𝑌 = (Base‘𝑇)
isghm.a	⊢ + = (+_g‘𝑆)
isghm.b	⊢ ⨣ = (+_g‘𝑇)

**Assertion**
Ref	Expression
isghm3	⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))

Distinct variable groups: 𝑣,𝑢,𝑆 𝑢,𝑇,𝑣 𝑢,𝑋,𝑣 𝑢, + ,𝑣 𝑢,𝑌,𝑣 𝑢, ⨣ ,𝑣 𝑢,𝐹,𝑣

**Proof of Theorem isghm3**
Step	Hyp	Ref	Expression
1		isghm.w	. . 3 ⊢ 𝑋 = (Base‘𝑆)
2		isghm.x	. . 3 ⊢ 𝑌 = (Base‘𝑇)
3		isghm.a	. . 3 ⊢ + = (+_g‘𝑆)
4		isghm.b	. . 3 ⊢ ⨣ = (+_g‘𝑇)
5	1, 2, 3, 4	isghm 19255	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
6	5	baib 535	1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +_gcplusg 17311 Grpcgrp 18973 GrpHom cghm 19252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-map 8886 df-ghm 19253

This theorem is referenced by: dfrhm2 20500

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