Theorem 0mhm 13236

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 19	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd))
2		eqid 2204	. . . . . 6 ⊢ (Base‘𝑁) = (Base‘𝑁)
3		0mhm.z	. . . . . 6 ⊢ 0 = (0g‘𝑁)
4	2, 3	mndidcl 13180	. . . . 5 ⊢ (𝑁 ∈ Mnd → 0 ∈ (Base‘𝑁))
5	4	adantl 277	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ (Base‘𝑁))
6		fconst6g 5468	. . . 4 ⊢ ( 0 ∈ (Base‘𝑁) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
7	5, 6	syl 14	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
8		simpr 110	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ Mnd)
9		eqid 2204	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
10	2, 9, 3	mndlid 13185	. . . . . . . 8 ⊢ ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → ( 0 (+g‘𝑁) 0 ) = 0 )
11	10	eqcomd 2210	. . . . . . 7 ⊢ ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → 0 = ( 0 (+g‘𝑁) 0 ))
12	8, 4, 11	syl2anc2 412	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 = ( 0 (+g‘𝑁) 0 ))
13	12	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 0 = ( 0 (+g‘𝑁) 0 ))
14		fn0g 13125	. . . . . . . . 9 ⊢ 0g Fn V
15	8	elexd 2784	. . . . . . . . 9 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ V)
16		funfvex 5587	. . . . . . . . . 10 ⊢ ((Fun 0g ∧ 𝑁 ∈ dom 0g) → (0g‘𝑁) ∈ V)
17	16	funfni 5370	. . . . . . . . 9 ⊢ ((0g Fn V ∧ 𝑁 ∈ V) → (0g‘𝑁) ∈ V)
18	14, 15, 17	sylancr 414	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g‘𝑁) ∈ V)
19	3, 18	eqeltrid 2291	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ V)
20	19	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 0 ∈ V)
21		0mhm.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
22		eqid 2204	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
23	21, 22	mndcl 13173	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
24	23	3expb 1206	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
25	24	adantlr 477	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
26		fvconst2g 5788	. . . . . 6 ⊢ (( 0 ∈ V ∧ (𝑥(+g‘𝑀)𝑦) ∈ 𝐵) → ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = 0 )
27	20, 25, 26	syl2anc 411	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = 0 )
28		simprl 529	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
29		fvconst2g 5788	. . . . . . 7 ⊢ (( 0 ∈ V ∧ 𝑥 ∈ 𝐵) → ((𝐵 × { 0 })‘𝑥) = 0 )
30	20, 28, 29	syl2anc 411	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘𝑥) = 0 )
31		simprr 531	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
32		fvconst2g 5788	. . . . . . 7 ⊢ (( 0 ∈ V ∧ 𝑦 ∈ 𝐵) → ((𝐵 × { 0 })‘𝑦) = 0 )
33	20, 31, 32	syl2anc 411	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘𝑦) = 0 )
34	30, 33	oveq12d 5952	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g‘𝑁) 0 ))
35	13, 27, 34	3eqtr4d 2247	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)))
36	35	ralrimivva 2587	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐵 × { 0 })‘(𝑥(+g‘𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g‘𝑁)((𝐵 × { 0 })‘𝑦)))
37		eqid 2204	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
38	21, 37	mndidcl 13180	. . . . 5 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
39	38	adantr 276	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g‘𝑀) ∈ 𝐵)
40		fvconst2g 5788	. . . 4 ⊢ (( 0 ∈ V ∧ (0g‘𝑀) ∈ 𝐵) → ((𝐵 × { 0 })‘(0g‘𝑀)) = 0 )
41	19, 39, 40	syl2anc 411	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 })‘(0g‘𝑀)) = 0 )
42	7, 36, 41	3jca 1179	. 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 ))
43	21, 2, 22, 9, 37, 3	ismhm 13211	. 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 )))
44	1, 42, 43	sylanbrc 417	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  ∀wral 2483  Vcvv 2771  {csn 3632   × cxp 4671   Fn wfn 5263  ⟶wf 5264  ‘cfv 5268  (class class class)co 5934  Basecbs 12751  +_gcplusg 12828  0_gc0g 13006  Mndcmnd 13166   MndHom cmhm 13207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-map 6727  df-inn 9019  df-2 9077  df-ndx 12754  df-slot 12755  df-base 12757  df-plusg 12841  df-0g 13008  df-mgm 13106  df-sgrp 13152  df-mnd 13167  df-mhm 13209

This theorem is referenced by:  0ghm  13512