Theorem mamulid 19235

Description: The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)

Hypotheses

Ref	Expression
mamumat1cl.b	|- B = ( Base ` R )
mamumat1cl.r	|- ( ph -> R e. Ring )
mamumat1cl.o	|- .1. = ( 1r ` R )
mamumat1cl.z	|- .0. = ( 0g ` R )
mamumat1cl.i	|- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
mamumat1cl.m	|- ( ph -> M e. Fin )
mamulid.n	|- ( ph -> N e. Fin )
mamulid.f	|- F = ( R maMul <. M , M , N >. )
mamulid.x	|- ( ph -> X e. ( B ^m ( M X. N ) ) )

Assertion

Ref	Expression
mamulid	|- ( ph -> ( I F X ) = X )

Distinct variable groups:   i, j, B   i, M, j   ph, i, j   .0. , i, j   .1. , i, j

Allowed substitution hints:   R( i, j)   F( i, j)   I( i, j)   N( i, j)   X( i, j)

Proof of Theorem mamulid

Dummy variables k l m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mamulid.f	. . . . 5 |- F = ( R maMul <. M , M , N >. )
2		mamumat1cl.b	. . . . 5 |- B = ( Base ` R )
3		eqid 2402	. . . . 5 |- ( .r ` R ) = ( .r ` R )
4		mamumat1cl.r	. . . . . 6 |- ( ph -> R e. Ring )
5	4	adantr 463	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> R e. Ring )
6		mamumat1cl.m	. . . . . 6 |- ( ph -> M e. Fin )
7	6	adantr 463	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> M e. Fin )
8		mamulid.n	. . . . . 6 |- ( ph -> N e. Fin )
9	8	adantr 463	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> N e. Fin )
10		mamumat1cl.o	. . . . . . 7 |- .1. = ( 1r ` R )
11		mamumat1cl.z	. . . . . . 7 |- .0. = ( 0g ` R )
12		mamumat1cl.i	. . . . . . 7 |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
13	2, 4, 10, 11, 12, 6	mamumat1cl 19233	. . . . . 6 |- ( ph -> I e. ( B ^m ( M X. M ) ) )
14	13	adantr 463	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> I e. ( B ^m ( M X. M ) ) )
15		mamulid.x	. . . . . 6 |- ( ph -> X e. ( B ^m ( M X. N ) ) )
16	15	adantr 463	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> X e. ( B ^m ( M X. N ) ) )
17		simprl 756	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> l e. M )
18		simprr 758	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> k e. N )
19	1, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18	mamufv 19181	. . . 4 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l ( I F X ) k ) = ( R gsumg ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) ) )
20		ringmnd 17527	. . . . . 6 |- ( R e. Ring -> R e. Mnd )
21	5, 20	syl 17	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> R e. Mnd )
22	4	ad2antrr 724	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> R e. Ring )
23		elmapi 7478	. . . . . . . . . 10 |- ( I e. ( B ^m ( M X. M ) ) -> I : ( M X. M ) --> B )
24	13, 23	syl 17	. . . . . . . . 9 |- ( ph -> I : ( M X. M ) --> B )
25	24	ad2antrr 724	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> I : ( M X. M ) --> B )
26		simplrl 762	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> l e. M )
27		simpr 459	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> m e. M )
28	25, 26, 27	fovrnd 6428	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( l I m ) e. B )
29		elmapi 7478	. . . . . . . . . 10 |- ( X e. ( B ^m ( M X. N ) ) -> X : ( M X. N ) --> B )
30	15, 29	syl 17	. . . . . . . . 9 |- ( ph -> X : ( M X. N ) --> B )
31	30	ad2antrr 724	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> X : ( M X. N ) --> B )
32		simplrr 763	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> k e. N )
33	31, 27, 32	fovrnd 6428	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( m X k ) e. B )
34	2, 3	ringcl 17532	. . . . . . 7 |- ( ( R e. Ring /\ ( l I m ) e. B /\ ( m X k ) e. B ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) e. B )
35	22, 28, 33, 34	syl3anc 1230	. . . . . 6 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) e. B )
36		eqid 2402	. . . . . 6 |- ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) = ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) )
37	35, 36	fmptd 6033	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) : M --> B )
38	26	3adant3 1017	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> l e. M )
39		simp2 998	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> m e. M )
40	2, 4, 10, 11, 12, 6	mat1comp 19234	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( l = m , .1. , .0. ) )
41		equcom 1818	. . . . . . . . . . . . 13 |- ( l = m <-> m = l )
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( l e. M /\ m e. M ) -> ( l = m <-> m = l ) )
43	42	ifbid 3907	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> if ( l = m , .1. , .0. ) = if ( m = l , .1. , .0. ) )
44	40, 43	eqtrd 2443	. . . . . . . . . 10 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
45	38, 39, 44	syl2anc 659	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
46		ifnefalse 3897	. . . . . . . . . 10 |- ( m =/= l -> if ( m = l , .1. , .0. ) = .0. )
47	46	3ad2ant3 1020	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> if ( m = l , .1. , .0. ) = .0. )
48	45, 47	eqtrd 2443	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = .0. )
49	48	oveq1d 6293	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) = ( .0. ( .r ` R ) ( m X k ) ) )
50	2, 3, 11	ringlz 17555	. . . . . . . . 9 |- ( ( R e. Ring /\ ( m X k ) e. B ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
51	22, 33, 50	syl2anc 659	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
52	51	3adant3 1017	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
53	49, 52	eqtrd 2443	. . . . . 6 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) = .0. )
54	53, 7	suppsssn 6938	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) supp .0. ) C_ { l } )
55	2, 11, 21, 7, 17, 37, 54	gsumpt 17309	. . . 4 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( R gsumg ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) ) = ( ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) ` l ) )
56		oveq2 6286	. . . . . . . 8 |- ( m = l -> ( l I m ) = ( l I l ) )
57		oveq1 6285	. . . . . . . 8 |- ( m = l -> ( m X k ) = ( l X k ) )
58	56, 57	oveq12d 6296	. . . . . . 7 |- ( m = l -> ( ( l I m ) ( .r ` R ) ( m X k ) ) = ( ( l I l ) ( .r ` R ) ( l X k ) ) )
59		ovex 6306	. . . . . . 7 |- ( ( l I l ) ( .r ` R ) ( l X k ) ) e. _V
60	58, 36, 59	fvmpt 5932	. . . . . 6 |- ( l e. M -> ( ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) ` l ) = ( ( l I l ) ( .r ` R ) ( l X k ) ) )
61	60	ad2antrl 726	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) ` l ) = ( ( l I l ) ( .r ` R ) ( l X k ) ) )
62		equequ1 1822	. . . . . . . . . 10 |- ( i = l -> ( i = j <-> l = j ) )
63	62	ifbid 3907	. . . . . . . . 9 |- ( i = l -> if ( i = j , .1. , .0. ) = if ( l = j , .1. , .0. ) )
64		equequ2 1823	. . . . . . . . . . 11 |- ( j = l -> ( l = j <-> l = l ) )
65	64	ifbid 3907	. . . . . . . . . 10 |- ( j = l -> if ( l = j , .1. , .0. ) = if ( l = l , .1. , .0. ) )
66		equid 1815	. . . . . . . . . . 11 |- l = l
67	66	iftruei 3892	. . . . . . . . . 10 |- if ( l = l , .1. , .0. ) = .1.
68	65, 67	syl6eq 2459	. . . . . . . . 9 |- ( j = l -> if ( l = j , .1. , .0. ) = .1. )
69		fvex 5859	. . . . . . . . . 10 |- ( 1r ` R ) e. _V
70	10, 69	eqeltri 2486	. . . . . . . . 9 |- .1. e. _V
71	63, 68, 12, 70	ovmpt2 6419	. . . . . . . 8 |- ( ( l e. M /\ l e. M ) -> ( l I l ) = .1. )
72	71	anidms 643	. . . . . . 7 |- ( l e. M -> ( l I l ) = .1. )
73	72	ad2antrl 726	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l I l ) = .1. )
74	73	oveq1d 6293	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( ( l I l ) ( .r ` R ) ( l X k ) ) = ( .1. ( .r ` R ) ( l X k ) ) )
75	30	fovrnda 6427	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l X k ) e. B )
76	2, 3, 10	ringlidm 17542	. . . . . 6 |- ( ( R e. Ring /\ ( l X k ) e. B ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
77	5, 75, 76	syl2anc 659	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
78	61, 74, 77	3eqtrd 2447	. . . 4 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) ` l ) = ( l X k ) )
79	19, 55, 78	3eqtrd 2447	. . 3 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l ( I F X ) k ) = ( l X k ) )
80	79	ralrimivva 2825	. 2 |- ( ph -> A. l e. M A. k e. N ( l ( I F X ) k ) = ( l X k ) )
81	2, 4, 1, 6, 6, 8, 13, 15	mamucl 19195	. . . . 5 |- ( ph -> ( I F X ) e. ( B ^m ( M X. N ) ) )
82		elmapi 7478	. . . . 5 |- ( ( I F X ) e. ( B ^m ( M X. N ) ) -> ( I F X ) : ( M X. N ) --> B )
83	81, 82	syl 17	. . . 4 |- ( ph -> ( I F X ) : ( M X. N ) --> B )
84		ffn 5714	. . . 4 |- ( ( I F X ) : ( M X. N ) --> B -> ( I F X ) Fn ( M X. N ) )
85	83, 84	syl 17	. . 3 |- ( ph -> ( I F X ) Fn ( M X. N ) )
86		ffn 5714	. . . 4 |- ( X : ( M X. N ) --> B -> X Fn ( M X. N ) )
87	30, 86	syl 17	. . 3 |- ( ph -> X Fn ( M X. N ) )
88		eqfnov2 6390	. . 3 |- ( ( ( I F X ) Fn ( M X. N ) /\ X Fn ( M X. N ) ) -> ( ( I F X ) = X <-> A. l e. M A. k e. N ( l ( I F X ) k ) = ( l X k ) ) )
89	85, 87, 88	syl2anc 659	. 2 |- ( ph -> ( ( I F X ) = X <-> A. l e. M A. k e. N ( l ( I F X ) k ) = ( l X k ) ) )
90	80, 89	mpbird 232	1 |- ( ph -> ( I F X ) = X )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2754   _Vcvv 3059   ifcif 3885   <.cotp 3980   |-> cmpt 4453   X. cxp 4821   Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   |-> cmpt2 6280   ^m cmap 7457   Fincfn 7554   Basecbs 14841   .rcmulr 14910   0gc0g 15054   gsum_g cgsu 15055   Mndcmnd 16243   1rcur 17473   Ringcrg 17518   maMul cmmul 19177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-gsum 15057  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-mamu 19178

This theorem is referenced by:  matring  19237  mat1  19241