Theorem c0mgm 20395

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 18666	. . 3 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
2	1	anim2i 617	. 2 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇)
4		c0mhm.0	. . . . . . 7 ⊢ 0 = (0g‘𝑇)
5	3, 4	mndidcl 18674	. . . . . 6 ⊢ (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
6	5	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
7	6	adantr 480	. . . 4 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ 𝑥 ∈ 𝐵) → 0 ∈ (Base‘𝑇))
8		c0mhm.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
9	7, 8	fmptd 7059	. . 3 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
10	5	ancli 548	. . . . . . . 8 ⊢ (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
11	10	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
12		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝑇) = (+g‘𝑇)
13	3, 12, 4	mndlid 18679	. . . . . . 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 480	. . . . 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 2737	. . . . . . 7 ⊢ ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
18		simprl 770	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
19	6	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 0 ∈ (Base‘𝑇))
20	16, 17, 18, 19	fvmptd 6948	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑎) = 0 )
21		eqidd 2737	. . . . . . 7 ⊢ ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
22		simprr 772	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
23	16, 21, 22, 19	fvmptd 6948	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑏) = 0 )
24	20, 23	oveq12d 7376	. . . . 5 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)) = ( 0 (+g‘𝑇) 0 ))
25		eqidd 2737	. . . . . 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 2736	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
28	26, 27	mgmcl 18568	. . . . . . . 8 ⊢ ((𝑆 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
29	28	3expb 1120	. . . . . . 7 ⊢ ((𝑆 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
30	29	adantlr 715	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
31	16, 25, 30, 19	fvmptd 6948	. . . . 5 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = 0 )
32	15, 24, 31	3eqtr4rd 2782	. . . 4 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))
33	32	ralrimivva 3179	. . 3 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))
34	9, 33	jca 511	. 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 18621	. 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 583	1 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ↦ cmpt 5179  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  0_gc0g 17359  Mgmcmgm 18563   MgmHom cmgmhm 18615  Mndcmnd 18659

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-0g 17361  df-mgm 18565  df-mgmhm 18617  df-sgrp 18644  df-mnd 18660

This theorem is referenced by:  c0rnghm  20468