Theorem 0mhm 13699

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 2232	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
3		0mhm.z	. . . . . 6 |- .0. = ( 0g ` N )
4	2, 3	mndidcl 13643	. . . . 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 5566	. . . 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 2232	. . . . . . . . 9 |- ( +g ` N ) = ( +g ` N )
10	2, 9, 3	mndlid 13648	. . . . . . . 8 |- ( ( N e. Mnd /\ .0. e. ( Base ` N ) ) -> ( .0. ( +g ` N ) .0. ) = .0. )
11	10	eqcomd 2238	. . . . . . 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 13588	. . . . . . . . 9 |- 0g Fn _V
15	8	elexd 2827	. . . . . . . . 9 |- ( ( M e. Mnd /\ N e. Mnd ) -> N e. _V )
16		funfvex 5687	. . . . . . . . . 10 |- ( ( Fun 0g /\ N e. dom 0g ) -> ( 0g ` N ) e. _V )
17	16	funfni 5458	. . . . . . . . 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 2319	. . . . . . 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 2232	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
23	21, 22	mndcl 13636	. . . . . . . 8 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
24	23	3expb 1231	. . . . . . 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 5898	. . . . . 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 5898	. . . . . . 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 5898	. . . . . . 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 6068	. . . . 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 2275	. . . 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 2624	. . 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 2232	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
38	21, 37	mndidcl 13643	. . . . 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 5898	. . . 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 1204	. 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 13674	. 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 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   {csn 3689   X. cxp 4747   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   Mndcmnd 13629   MndHom cmhm 13670

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mhm 13672

This theorem is referenced by:  0ghm  13975