c0ghm - Metamath Proof Explorer

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Theorem c0ghm 20510

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 18982	. . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
2		grpmnd 18982	. . . 4 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
3	1, 2	anim12i 622	. . 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 20509	. . 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 19267	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
10	9	eleq2d 2848	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇)))
11	8, 10	mpbird 259	1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ↦ cmpt 5181 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 0_gc0g 17468 Mndcmnd 18768 MndHom cmhm 18815 Grpcgrp 18975 GrpHom cghm 19253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-map 8810 df-0g 17470 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-mhm 18817 df-grp 18978 df-ghm 19254

This theorem is referenced by: c0rhm 20584 c0rnghm 20585

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