isghm3 - Metamath Proof Explorer

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Theorem isghm3 19115

Description: Property of a group homomorphism, similar to ismhm 18678. (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 19113	. 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 1540 ∈ wcel 2109 ∀wral 3044 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 Basecbs 17139 +_gcplusg 17180 Grpcgrp 18831 GrpHom cghm 19110

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fv 6494 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-map 8762 df-ghm 19111

This theorem is referenced by: dfrhm2 20378

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