Theorem c0mhm 20379

Description: The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
c0mhm	⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))

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

Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem c0mhm

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

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
2		c0mhm.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑇)
3	1, 2	mndidcl 18657	. . . . . . 7 ⊢ (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
4	3	adantl 481	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
5	4	adantr 480	. . . . 5 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 ∈ 𝐵) → 0 ∈ (Base‘𝑇))
6		c0mhm.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )
7	5, 6	fmptd 7047	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
8	3	ancli 548	. . . . . . . . 9 ⊢ (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
10		eqid 2731	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
11	1, 10, 2	mndlid 18662	. . . . . . . 8 ⊢ ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g‘𝑇) 0 ) = 0 )
12	9, 11	syl 17	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ( 0 (+g‘𝑇) 0 ) = 0 )
13	12	adantr 480	. . . . . 6 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 0 (+g‘𝑇) 0 ) = 0 )
14	6	a1i 11	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ))
15		eqidd 2732	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
16		simprl 770	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
17	4	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 0 ∈ (Base‘𝑇))
18	14, 15, 16, 17	fvmptd 6936	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑎) = 0 )
19		eqidd 2732	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
20		simprr 772	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
21	14, 19, 20, 17	fvmptd 6936	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑏) = 0 )
22	18, 21	oveq12d 7364	. . . . . 6 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)) = ( 0 (+g‘𝑇) 0 ))
23		eqidd 2732	. . . . . . 7 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = (𝑎(+g‘𝑆)𝑏)) → 0 = 0 )
24		c0mhm.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
25		eqid 2731	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
26	24, 25	mndcl 18650	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
27	26	3expb 1120	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
28	27	adantlr 715	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
29	14, 23, 28, 17	fvmptd 6936	. . . . . 6 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = 0 )
30	13, 22, 29	3eqtr4rd 2777	. . . . 5 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))
31	30	ralrimivva 3175	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))
32	6	a1i 11	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ))
33		eqidd 2732	. . . . 5 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 = (0g‘𝑆)) → 0 = 0 )
34		eqid 2731	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
35	24, 34	mndidcl 18657	. . . . . 6 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ 𝐵)
36	35	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (0g‘𝑆) ∈ 𝐵)
37	32, 33, 36, 4	fvmptd 6936	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻‘(0g‘𝑆)) = 0 )
38	7, 31, 37	3jca 1128	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)) ∧ (𝐻‘(0g‘𝑆)) = 0 ))
39	38	ancli 548	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)) ∧ (𝐻‘(0g‘𝑆)) = 0 )))
40	24, 1, 25, 10, 34, 2	ismhm 18693	. 2 ⊢ (𝐻 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)) ∧ (𝐻‘(0g‘𝑆)) = 0 )))
41	39, 40	sylibr 234	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ↦ cmpt 5172  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  +_gcplusg 17161  0_gc0g 17343  Mndcmnd 18642   MndHom cmhm 18689

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-0g 17345  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691

This theorem is referenced by:  c0ghm  20380  c0rhm  20450