Theorem 0mhm 12873

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 2177	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
3		0mhm.z	. . . . . 6 |- .0. = ( 0g ` N )
4	2, 3	mndidcl 12831	. . . . 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 5415	. . . 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 2177	. . . . . . . . 9 |- ( +g ` N ) = ( +g ` N )
10	2, 9, 3	mndlid 12836	. . . . . . . 8 |- ( ( N e. Mnd /\ .0. e. ( Base ` N ) ) -> ( .0. ( +g ` N ) .0. ) = .0. )
11	10	eqcomd 2183	. . . . . . 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 12794	. . . . . . . . 9 |- 0g Fn _V
15	8	elexd 2751	. . . . . . . . 9 |- ( ( M e. Mnd /\ N e. Mnd ) -> N e. _V )
16		funfvex 5533	. . . . . . . . . 10 |- ( ( Fun 0g /\ N e. dom 0g ) -> ( 0g ` N ) e. _V )
17	16	funfni 5317	. . . . . . . . 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 2264	. . . . . . 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 2177	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
23	21, 22	mndcl 12824	. . . . . . . 8 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
24	23	3expb 1204	. . . . . . 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 5731	. . . . . 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 5731	. . . . . . 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 5731	. . . . . . 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 5893	. . . . 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 2220	. . . 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 2559	. . 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 2177	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
38	21, 37	mndidcl 12831	. . . . 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 5731	. . . 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 1177	. 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 12853	. 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2738   {csn 3593   X. cxp 4625   Fn wfn 5212   -->wf 5213   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   0gc0g 12705   Mndcmnd 12817   MndHom cmhm 12849

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-inn 8920  df-2 8978  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-mhm 12851

This theorem is referenced by: (None)