Theorem 0mhm 13318

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 2205	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
3		0mhm.z	. . . . . 6 |- .0. = ( 0g ` N )
4	2, 3	mndidcl 13262	. . . . 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 5474	. . . 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 2205	. . . . . . . . 9 |- ( +g ` N ) = ( +g ` N )
10	2, 9, 3	mndlid 13267	. . . . . . . 8 |- ( ( N e. Mnd /\ .0. e. ( Base ` N ) ) -> ( .0. ( +g ` N ) .0. ) = .0. )
11	10	eqcomd 2211	. . . . . . 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 13207	. . . . . . . . 9 |- 0g Fn _V
15	8	elexd 2785	. . . . . . . . 9 |- ( ( M e. Mnd /\ N e. Mnd ) -> N e. _V )
16		funfvex 5593	. . . . . . . . . 10 |- ( ( Fun 0g /\ N e. dom 0g ) -> ( 0g ` N ) e. _V )
17	16	funfni 5376	. . . . . . . . 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 2292	. . . . . . 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 2205	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
23	21, 22	mndcl 13255	. . . . . . . 8 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
24	23	3expb 1207	. . . . . . 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 5798	. . . . . 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 5798	. . . . . . 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 5798	. . . . . . 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 5962	. . . . 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 2248	. . . 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 2588	. . 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 2205	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
38	21, 37	mndidcl 13262	. . . . 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 5798	. . . 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 1180	. 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 13293	. 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 981   = wceq 1373   e. wcel 2176   A.wral 2484   _Vcvv 2772   {csn 3633   X. cxp 4673   Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   0gc0g 13088   Mndcmnd 13248   MndHom cmhm 13289

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-mhm 13291

This theorem is referenced by:  0ghm  13594