Theorem c0mgm 42924

Description: The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
c0mgm	⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑆   𝑥,𝑇   𝑥, 0

Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem c0mgm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndmgm 17686	. . 3 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
2	1	anim2i 610	. 2 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3		eqid 2778	. . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇)
4		c0mhm.0	. . . . . . 7 ⊢ 0 = (0g‘𝑇)
5	3, 4	mndidcl 17694	. . . . . 6 ⊢ (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
6	5	adantl 475	. . . . 5 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
7	6	adantr 474	. . . 4 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ 𝑥 ∈ 𝐵) → 0 ∈ (Base‘𝑇))
8		c0mhm.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
9	7, 8	fmptd 6648	. . 3 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
10	5	ancli 544	. . . . . . . 8 ⊢ (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
11	10	adantl 475	. . . . . . 7 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
12		eqid 2778	. . . . . . . 8 ⊢ (+g‘𝑇) = (+g‘𝑇)
13	3, 12, 4	mndlid 17697	. . . . . . 7 ⊢ ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g‘𝑇) 0 ) = 0 )
14	11, 13	syl 17	. . . . . 6 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ( 0 (+g‘𝑇) 0 ) = 0 )
15	14	adantr 474	. . . . 5 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 0 (+g‘𝑇) 0 ) = 0 )
16	8	a1i 11	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ))
17		eqidd 2779	. . . . . . 7 ⊢ ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
18		simprl 761	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
19	6	adantr 474	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 0 ∈ (Base‘𝑇))
20	16, 17, 18, 19	fvmptd 6548	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑎) = 0 )
21		eqidd 2779	. . . . . . 7 ⊢ ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
22		simprr 763	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
23	16, 21, 22, 19	fvmptd 6548	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑏) = 0 )
24	20, 23	oveq12d 6940	. . . . 5 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)) = ( 0 (+g‘𝑇) 0 ))
25		eqidd 2779	. . . . . 6 ⊢ ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = (𝑎(+g‘𝑆)𝑏)) → 0 = 0 )
26		c0mhm.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑆)
27		eqid 2778	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
28	26, 27	mgmcl 17631	. . . . . . . 8 ⊢ ((𝑆 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
29	28	3expb 1110	. . . . . . 7 ⊢ ((𝑆 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
30	29	adantlr 705	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
31	16, 25, 30, 19	fvmptd 6548	. . . . 5 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = 0 )
32	15, 24, 31	3eqtr4rd 2825	. . . 4 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))
33	32	ralrimivva 3153	. . 3 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))
34	9, 33	jca 507	. 2 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏))))
35	26, 3, 27, 12	ismgmhm 42798	. 2 ⊢ (𝐻 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))))
36	2, 34, 35	sylanbrc 578	1 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ↦ cmpt 4965  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  +_gcplusg 16338  0_gc0g 16486  Mgmcmgm 17626  Mndcmnd 17680   MgmHom cmgmhm 42792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-mgmhm 42794

This theorem is referenced by:  c0rnghm  42928