c0ghm - Metamath Proof Explorer

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Theorem c0ghm 20742

Description: The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)

**Hypotheses**
Ref	Expression
c0mhm.b	⊢ 𝐵 = (Base‘𝑆)
c0mhm.0	⊢ 0 = (0_g‘𝑇)
c0mhm.h	⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )

**Assertion**
Ref	Expression
c0ghm	⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑆 𝑥,𝑇 𝑥, 0 Allowed substitution hint: 𝐻(𝑥)

**Proof of Theorem c0ghm**
Step	Hyp	Ref	Expression
1		grpmnd 19201	. . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
2		grpmnd 19201	. . . 4 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
3	1, 2	anim12i 625	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
4		c0mhm.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
5		c0mhm.0	. . . 4 ⊢ 0 = (0_g‘𝑇)
6		c0mhm.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
7	4, 5, 6	c0mhm 20741	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
8	3, 7	syl 18	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 MndHom 𝑇))
9		ghmmhmb 19491	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
10	9	eleq2d 2850	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇)))
11	8, 10	mpbird 260	1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ↦ cmpt 5192 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 0_gc0g 17660 Mndcmnd 18973 MndHom cmhm 19026 Grpcgrp 19194 GrpHom cghm 19477

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-map 8887 df-0g 17662 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-grp 19197 df-ghm 19478

This theorem is used by: c0rhm 20837 c0rnghm 20838

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