Theorem 0mhm 18832

Description: The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
0mhm.z	⊢ 0 = (0g‘𝑁)
0mhm.b	⊢ 𝐵 = (Base‘𝑀)

Assertion

Ref	Expression
0mhm	⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))

Proof of Theorem 0mhm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd))
2		eqid 2737	. . . . . 6 ⊢ (Base‘𝑁) = (Base‘𝑁)
3		0mhm.z	. . . . . 6 ⊢ 0 = (0g‘𝑁)
4	2, 3	mndidcl 18762	. . . . 5 ⊢ (𝑁 ∈ Mnd → 0 ∈ (Base‘𝑁))
5	4	adantl 481	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ (Base‘𝑁))
6		fconst6g 6797	. . . 4 ⊢ ( 0 ∈ (Base‘𝑁) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
7	5, 6	syl 17	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
8		simpr 484	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ Mnd)
9		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
10	2, 9, 3	mndlid 18767	. . . . . . . 8 ⊢ ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → ( 0 (+g‘𝑁) 0 ) = 0 )
11	10	eqcomd 2743	. . . . . . 7 ⊢ ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → 0 = ( 0 (+g‘𝑁) 0 ))
12	8, 4, 11	syl2anc2 585	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 = ( 0 (+g‘𝑁) 0 ))
13	12	adantr 480	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 0 = ( 0 (+g‘𝑁) 0 ))
14		0mhm.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
15		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	14, 15	mndcl 18755	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
17	16	3expb 1121	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
18	17	adantlr 715	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
19	3	fvexi 6920	. . . . . . 7 ⊢ 0 ∈ V
20	19	fvconst2 7224	. . . . . 6 ⊢ ((𝑥(+g‘𝑀)𝑦) ∈ 𝐵 → ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = 0 )
21	18, 20	syl 17	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = 0 )
22	19	fvconst2 7224	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ((𝐵 × { 0 })‘𝑥) = 0 )
23	19	fvconst2 7224	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 )
24	22, 23	oveqan12d 7450	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g‘𝑁) 0 ))
25	24	adantl 481	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g‘𝑁) 0 ))
26	13, 21, 25	3eqtr4d 2787	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)))
27	26	ralrimivva 3202	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)))
28		eqid 2737	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
29	14, 28	mndidcl 18762	. . . . 5 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
30	29	adantr 480	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g‘𝑀) ∈ 𝐵)
31	19	fvconst2 7224	. . . 4 ⊢ ((0g‘𝑀) ∈ 𝐵 → ((𝐵 × { 0 })‘(0g‘𝑀)) = 0 )
32	30, 31	syl 17	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 })‘(0g‘𝑀)) = 0 )
33	7, 27, 32	3jca 1129	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 }):𝐵⟶(Base‘𝑁) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)) ∧ ((𝐵 × { 0 })‘(0g‘𝑀)) = 0 ))
34	14, 2, 15, 9, 28, 3	ismhm 18798	. 2 ⊢ ((𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁) ↔ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ ((𝐵 × { 0 }):𝐵⟶(Base‘𝑁) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)) ∧ ((𝐵 × { 0 })‘(0g‘𝑀)) = 0 )))
35	1, 33, 34	sylanbrc 583	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {csn 4626   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  0_gc0g 17484  Mndcmnd 18747   MndHom cmhm 18794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796

This theorem is referenced by:  0ghm  19248