c0ghm - Metamath Proof Explorer

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Theorem c0ghm 20395

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 18868	. . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
2		grpmnd 18868	. . . 4 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
3	1, 2	anim12i 613	. . 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 20394	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
8	3, 7	syl 17	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 MndHom 𝑇))
9		ghmmhmb 19154	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
10	9	eleq2d 2820	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇)))
11	8, 10	mpbird 257	1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ↦ cmpt 5177 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 0_gc0g 17357 Mndcmnd 18657 MndHom cmhm 18704 Grpcgrp 18861 GrpHom cghm 19139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-map 8763 df-0g 17359 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-grp 18864 df-ghm 19140

This theorem is referenced by: c0rhm 20465 c0rnghm 20466

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