Theorem c0mhm 41700

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 2610	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
2		c0mhm.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑇)
3	1, 2	mndidcl 17131	. . . . . . 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 6292	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
8	3	ancli 572	. . . . . . . . 9 ⊢ (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
10		eqid 2610	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
11	1, 10, 2	mndlid 17134	. . . . . . . 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 2611	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
16		simprl 790	. . . . . . . 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 6197	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑎) = 0 )
19		eqidd 2611	. . . . . . . 8 ⊢ ((((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
20		simprr 792	. . . . . . . 8 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
21	14, 19, 20, 17	fvmptd 6197	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘𝑏) = 0 )
22	18, 21	oveq12d 6567	. . . . . 6 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)) = ( 0 (+g‘𝑇) 0 ))
23		eqidd 2611	. . . . . . 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 2610	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
26	24, 25	mndcl 17124	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
27	26	3expb 1258	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
28	27	adantlr 747	. . . . . . 7 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝐵)
29	14, 23, 28, 17	fvmptd 6197	. . . . . 6 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = 0 )
30	13, 22, 29	3eqtr4rd 2655	. . . . 5 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐻‘(𝑎(+g‘𝑆)𝑏)) = ((𝐻‘𝑎)(+g‘𝑇)(𝐻‘𝑏)))
31	30	ralrimivva 2954	. . . 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 2611	. . . . 5 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ 𝑥 = (0g‘𝑆)) → 0 = 0 )
34		eqid 2610	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
35	24, 34	mndidcl 17131	. . . . . 6 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ 𝐵)
36	35	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (0g‘𝑆) ∈ 𝐵)
37	32, 33, 36, 4	fvmptd 6197	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐻‘(0g‘𝑆)) = 0 )
38	7, 31, 37	3jca 1235	. . 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 572	. 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 17160	. 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 223	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Mndcmnd 17117   MndHom cmhm 17156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158

This theorem is referenced by:  c0ghm  41701  c0rhm  41702