Theorem 0mhm 13568

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 2231	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
3		0mhm.z	. . . . . 6 |- .0. = ( 0g ` N )
4	2, 3	mndidcl 13512	. . . . 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 5535	. . . 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 2231	. . . . . . . . 9 |- ( +g ` N ) = ( +g ` N )
10	2, 9, 3	mndlid 13517	. . . . . . . 8 |- ( ( N e. Mnd /\ .0. e. ( Base ` N ) ) -> ( .0. ( +g ` N ) .0. ) = .0. )
11	10	eqcomd 2237	. . . . . . 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 13457	. . . . . . . . 9 |- 0g Fn _V
15	8	elexd 2816	. . . . . . . . 9 |- ( ( M e. Mnd /\ N e. Mnd ) -> N e. _V )
16		funfvex 5656	. . . . . . . . . 10 |- ( ( Fun 0g /\ N e. dom 0g ) -> ( 0g ` N ) e. _V )
17	16	funfni 5432	. . . . . . . . 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 2318	. . . . . . 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 2231	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
23	21, 22	mndcl 13505	. . . . . . . 8 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
24	23	3expb 1230	. . . . . . 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 5867	. . . . . 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 531	. . . . . . 7 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
29		fvconst2g 5867	. . . . . . 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 533	. . . . . . 7 |- ( ( ( M e. Mnd /\ N e. Mnd ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
32		fvconst2g 5867	. . . . . . 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 6035	. . . . 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 2274	. . . 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 2614	. . 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 2231	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
38	21, 37	mndidcl 13512	. . . . 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 5867	. . . 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 1203	. 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 13543	. 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   {csn 3669   X. cxp 4723   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Mndcmnd 13498   MndHom cmhm 13539

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mhm 13541

This theorem is referenced by:  0ghm  13844