Theorem 0mhm 12968

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 e. Mnd /\ N e. Mnd ) -> ( B X. { .0. } ) e. ( 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 e. Mnd /\ N e. Mnd ) -> ( M e. Mnd /\ N e. Mnd ) )
2		eqid 2189	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
3		0mhm.z	. . . . . 6 |- .0. = ( 0g ` N )
4	2, 3	mndidcl 12921	. . . . 5 |- ( N e. Mnd -> .0. e. ( Base ` N ) )
5	4	adantl 277	. . . 4 |- ( ( M e. Mnd /\ N e. Mnd ) -> .0. e. ( Base ` N ) )
6		fconst6g 5440	. . . 4 |- ( .0. e. ( Base ` N ) -> ( B X. { .0. } ) : B --> ( Base ` N ) )
7	5, 6	syl 14	. . 3 |- ( ( M e. Mnd /\ N e. Mnd ) -> ( B X. { .0. } ) : B --> ( Base ` N ) )
8		simpr 110	. . . . . . 7 |- ( ( M e. Mnd /\ N e. Mnd ) -> N e. Mnd )
9		eqid 2189	. . . . . . . . 9 |- ( +g ` N ) = ( +g ` N )
10	2, 9, 3	mndlid 12926	. . . . . . . 8 |- ( ( N e. Mnd /\ .0. e. ( Base ` N ) ) -> ( .0. ( +g ` N ) .0. ) = .0. )
11	10	eqcomd 2195	. . . . . . 7 |- ( ( N e. Mnd /\ .0. e. ( Base ` N ) ) -> .0. = ( .0. ( +g ` N ) .0. ) )
12	8, 4, 11	syl2anc2 412	. . . . . 6 |- ( ( M e. Mnd /\ N e. Mnd ) -> .0. = ( .0. ( +g ` N ) .0. ) )
13	12	adantr 276	. . . . 5 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> .0. = ( .0. ( +g ` N ) .0. ) )
14		fn0g 12869	. . . . . . . . 9 |- 0g Fn _V
15	8	elexd 2769	. . . . . . . . 9 |- ( ( M e. Mnd /\ N e. Mnd ) -> N e. _V )
16		funfvex 5558	. . . . . . . . . 10 |- ( ( Fun 0g /\ N e. dom 0g ) -> ( 0g ` N ) e. _V )
17	16	funfni 5342	. . . . . . . . 9 |- ( ( 0g Fn _V /\ N e. _V ) -> ( 0g ` N ) e. _V )
18	14, 15, 17	sylancr 414	. . . . . . . 8 |- ( ( M e. Mnd /\ N e. Mnd ) -> ( 0g ` N ) e. _V )
19	3, 18	eqeltrid 2276	. . . . . . 7 |- ( ( M e. Mnd /\ N e. Mnd ) -> .0. e. _V )
20	19	adantr 276	. . . . . 6 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> .0. e. _V )
21		0mhm.b	. . . . . . . . 9 |- B = ( Base ` M )
22		eqid 2189	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
23	21, 22	mndcl 12914	. . . . . . . 8 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
24	23	3expb 1206	. . . . . . 7 |- ( ( M e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` M ) y ) e. B )
25	24	adantlr 477	. . . . . 6 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` M ) y ) e. B )
26		fvconst2g 5758	. . . . . 6 |- ( ( .0. e. _V /\ ( x ( +g ` M ) y ) e. B ) -> ( ( B X. { .0. } ) ` ( x ( +g ` M ) y ) ) = .0. )
27	20, 25, 26	syl2anc 411	. . . . 5 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> ( ( B X. { .0. } ) ` ( x ( +g ` M ) y ) ) = .0. )
28		simprl 529	. . . . . . 7 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
29		fvconst2g 5758	. . . . . . 7 |- ( ( .0. e. _V /\ x e. B ) -> ( ( B X. { .0. } ) ` x ) = .0. )
30	20, 28, 29	syl2anc 411	. . . . . 6 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> ( ( B X. { .0. } ) ` x ) = .0. )
31		simprr 531	. . . . . . 7 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
32		fvconst2g 5758	. . . . . . 7 |- ( ( .0. e. _V /\ y e. B ) -> ( ( B X. { .0. } ) ` y ) = .0. )
33	20, 31, 32	syl2anc 411	. . . . . 6 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> ( ( B X. { .0. } ) ` y ) = .0. )
34	30, 33	oveq12d 5922	. . . . 5 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> ( ( ( B X. { .0. } ) ` x ) ( +g ` N ) ( ( B X. { .0. } ) ` y ) ) = ( .0. ( +g ` N ) .0. ) )
35	13, 27, 34	3eqtr4d 2232	. . . 4 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> ( ( B X. { .0. } ) ` ( x ( +g ` M ) y ) ) = ( ( ( B X. { .0. } ) ` x ) ( +g ` N ) ( ( B X. { .0. } ) ` y ) ) )
36	35	ralrimivva 2572	. . 3 |- ( ( M e. Mnd /\ N e. Mnd ) -> A. x e. B A. y e. B ( ( B X. { .0. } ) ` ( x ( +g ` M ) y ) ) = ( ( ( B X. { .0. } ) ` x ) ( +g ` N ) ( ( B X. { .0. } ) ` y ) ) )
37		eqid 2189	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
38	21, 37	mndidcl 12921	. . . . 5 |- ( M e. Mnd -> ( 0g ` M ) e. B )
39	38	adantr 276	. . . 4 |- ( ( M e. Mnd /\ N e. Mnd ) -> ( 0g ` M ) e. B )
40		fvconst2g 5758	. . . 4 |- ( ( .0. e. _V /\ ( 0g ` M ) e. B ) -> ( ( B X. { .0. } ) ` ( 0g ` M ) ) = .0. )
41	19, 39, 40	syl2anc 411	. . 3 |- ( ( M e. Mnd /\ N e. Mnd ) -> ( ( B X. { .0. } ) ` ( 0g ` M ) ) = .0. )
42	7, 36, 41	3jca 1179	. 2 |- ( ( M e. Mnd /\ N e. Mnd ) -> ( ( B X. { .0. } ) : B --> ( Base ` N ) /\ A. x e. B A. y e. B ( ( B X. { .0. } ) ` ( x ( +g ` M ) y ) ) = ( ( ( B X. { .0. } ) ` x ) ( +g ` N ) ( ( B X. { .0. } ) ` y ) ) /\ ( ( B X. { .0. } ) ` ( 0g ` M ) ) = .0. ) )
43	21, 2, 22, 9, 37, 3	ismhm 12943	. 2 |- ( ( B X. { .0. } ) e. ( M MndHom N ) <-> ( ( M e. Mnd /\ N e. Mnd ) /\ ( ( B X. { .0. } ) : B --> ( Base ` N ) /\ A. x e. B A. y e. B ( ( B X. { .0. } ) ` ( x ( +g ` M ) y ) ) = ( ( ( B X. { .0. } ) ` x ) ( +g ` N ) ( ( B X. { .0. } ) ` y ) ) /\ ( ( B X. { .0. } ) ` ( 0g ` M ) ) = .0. ) ) )
44	1, 42, 43	sylanbrc 417	1 |- ( ( M e. Mnd /\ N e. Mnd ) -> ( B X. { .0. } ) e. ( M MndHom N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   _Vcvv 2756   {csn 3614   X. cxp 4649   Fn wfn 5237   -->wf 5238   ` cfv 5242  (class class class)co 5904   Basecbs 12529   +g cplusg 12606   0gc0g 12778   Mndcmnd 12907   MndHom cmhm 12939

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-cnex 7942  ax-resscn 7943  ax-1re 7945  ax-addrcl 7948

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-fv 5250  df-riota 5859  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-map 6684  df-inn 8961  df-2 9019  df-ndx 12532  df-slot 12533  df-base 12535  df-plusg 12619  df-0g 12780  df-mgm 12850  df-sgrp 12895  df-mnd 12908  df-mhm 12941

This theorem is referenced by:  0ghm  13230