Theorem c0mgm 20439

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 18709	. . 3 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
2	1	anim2i 618	. 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 18717	. . . . . 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 7066	. . 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 18722	. . . . . . 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 771	. . . . . . 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 6955	. . . . . 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 773	. . . . . . 7 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
23	16, 21, 22, 19	fvmptd 6955	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑏) = 0 )
24	20, 23	oveq12d 7385	. . . . 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 18611	. . . . . . . 8 ⊢ ((𝑆 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
29	28	3expb 1121	. . . . . . 7 ⊢ ((𝑆 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
30	29	adantlr 716	. . . . . 6 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
31	16, 25, 30, 19	fvmptd 6955	. . . . 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 3180	. . 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 18664	. 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 584	1 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Mgmcmgm 18606   MgmHom cmgmhm 18658  Mndcmnd 18702

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-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-map 8775  df-0g 17404  df-mgm 18608  df-mgmhm 18660  df-sgrp 18687  df-mnd 18703

This theorem is referenced by:  c0rnghm  20512