Theorem 0mhm 18630

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 2736	. . . . . 6 ⊢ (Base‘𝑁) = (Base‘𝑁)
3		0mhm.z	. . . . . 6 ⊢ 0 = (0g‘𝑁)
4	2, 3	mndidcl 18571	. . . . 5 ⊢ (𝑁 ∈ Mnd → 0 ∈ (Base‘𝑁))
5	4	adantl 482	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ (Base‘𝑁))
6		fconst6g 6731	. . . 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 485	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ Mnd)
9		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
10	2, 9, 3	mndlid 18576	. . . . . . . 8 ⊢ ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → ( 0 (+g‘𝑁) 0 ) = 0 )
11	10	eqcomd 2742	. . . . . . 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 481	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 0 = ( 0 (+g‘𝑁) 0 ))
14		0mhm.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
15		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	14, 15	mndcl 18564	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
17	16	3expb 1120	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
18	17	adantlr 713	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
19	3	fvexi 6856	. . . . . . 7 ⊢ 0 ∈ V
20	19	fvconst2 7153	. . . . . 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 7153	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ((𝐵 × { 0 })‘𝑥) = 0 )
23	19	fvconst2 7153	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 )
24	22, 23	oveqan12d 7376	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g‘𝑁) 0 ))
25	24	adantl 482	. . . . 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 2786	. . . 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 3197	. . 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 2736	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
29	14, 28	mndidcl 18571	. . . . 5 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
30	29	adantr 481	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g‘𝑀) ∈ 𝐵)
31	19	fvconst2 7153	. . . 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 1128	. 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 18603	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {csn 4586   × cxp 5631  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321  Mndcmnd 18556   MndHom cmhm 18599

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601

This theorem is referenced by:  0ghm  19022