c0ghm - Metamath Proof Explorer

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Theorem c0ghm 20441

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 18916	. . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
2		grpmnd 18916	. . . 4 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
3	1, 2	anim12i 614	. . 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 20440	. . 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 19202	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
10	9	eleq2d 2822	. 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 1542 ∈ wcel 2114 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 0_gc0g 17402 Mndcmnd 18702 MndHom cmhm 18749 Grpcgrp 18909 GrpHom cghm 19187

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-map 8775 df-0g 17404 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-grp 18912 df-ghm 19188

This theorem is referenced by: c0rhm 20511 c0rnghm 20512

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