Theorem mamulid 19521

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 2462	. . . . 5 |- ( .r ` R ) = ( .r ` R )
4		mamumat1cl.r	. . . . . 6 |- ( ph -> R e. Ring )
5	4	adantr 471	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> R e. Ring )
6		mamumat1cl.m	. . . . . 6 |- ( ph -> M e. Fin )
7	6	adantr 471	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> M e. Fin )
8		mamulid.n	. . . . . 6 |- ( ph -> N e. Fin )
9	8	adantr 471	. . . . 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 19519	. . . . . 6 |- ( ph -> I e. ( B ^m ( M X. M ) ) )
14	13	adantr 471	. . . . 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 471	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> X e. ( B ^m ( M X. N ) ) )
17		simprl 769	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> l e. M )
18		simprr 771	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> k e. N )
19	1, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18	mamufv 19467	. . . 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 17844	. . . . . 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 737	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> R e. Ring )
23		elmapi 7524	. . . . . . . . . 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 737	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> I : ( M X. M ) --> B )
26		simplrl 775	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> l e. M )
27		simpr 467	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> m e. M )
28	25, 26, 27	fovrnd 6473	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( l I m ) e. B )
29		elmapi 7524	. . . . . . . . . 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 737	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> X : ( M X. N ) --> B )
32		simplrr 776	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> k e. N )
33	31, 27, 32	fovrnd 6473	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( m X k ) e. B )
34	2, 3	ringcl 17849	. . . . . . 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 1276	. . . . . 6 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) e. B )
36		eqid 2462	. . . . . 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 6074	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) : M --> B )
38	26	3adant3 1034	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> l e. M )
39		simp2 1015	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> m e. M )
40	2, 4, 10, 11, 12, 6	mat1comp 19520	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( l = m , .1. , .0. ) )
41		equcom 1873	. . . . . . . . . . . . 13 |- ( l = m <-> m = l )
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( l e. M /\ m e. M ) -> ( l = m <-> m = l ) )
43	42	ifbid 3915	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> if ( l = m , .1. , .0. ) = if ( m = l , .1. , .0. ) )
44	40, 43	eqtrd 2496	. . . . . . . . . 10 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
45	38, 39, 44	syl2anc 671	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
46		ifnefalse 3905	. . . . . . . . . 10 |- ( m =/= l -> if ( m = l , .1. , .0. ) = .0. )
47	46	3ad2ant3 1037	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> if ( m = l , .1. , .0. ) = .0. )
48	45, 47	eqtrd 2496	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = .0. )
49	48	oveq1d 6335	. . . . . . 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 17872	. . . . . . . . 9 |- ( ( R e. Ring /\ ( m X k ) e. B ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
51	22, 33, 50	syl2anc 671	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
52	51	3adant3 1034	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
53	49, 52	eqtrd 2496	. . . . . 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 6982	. . . . 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 17649	. . . 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 6328	. . . . . . . 8 |- ( m = l -> ( l I m ) = ( l I l ) )
57		oveq1 6327	. . . . . . . 8 |- ( m = l -> ( m X k ) = ( l X k ) )
58	56, 57	oveq12d 6338	. . . . . . 7 |- ( m = l -> ( ( l I m ) ( .r ` R ) ( m X k ) ) = ( ( l I l ) ( .r ` R ) ( l X k ) ) )
59		ovex 6348	. . . . . . 7 |- ( ( l I l ) ( .r ` R ) ( l X k ) ) e. _V
60	58, 36, 59	fvmpt 5976	. . . . . 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 739	. . . . 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 1878	. . . . . . . . . 10 |- ( i = l -> ( i = j <-> l = j ) )
63	62	ifbid 3915	. . . . . . . . 9 |- ( i = l -> if ( i = j , .1. , .0. ) = if ( l = j , .1. , .0. ) )
64		equequ2 1879	. . . . . . . . . . 11 |- ( j = l -> ( l = j <-> l = l ) )
65	64	ifbid 3915	. . . . . . . . . 10 |- ( j = l -> if ( l = j , .1. , .0. ) = if ( l = l , .1. , .0. ) )
66		equid 1866	. . . . . . . . . . 11 |- l = l
67	66	iftruei 3900	. . . . . . . . . 10 |- if ( l = l , .1. , .0. ) = .1.
68	65, 67	syl6eq 2512	. . . . . . . . 9 |- ( j = l -> if ( l = j , .1. , .0. ) = .1. )
69		fvex 5902	. . . . . . . . . 10 |- ( 1r ` R ) e. _V
70	10, 69	eqeltri 2536	. . . . . . . . 9 |- .1. e. _V
71	63, 68, 12, 70	ovmpt2 6464	. . . . . . . 8 |- ( ( l e. M /\ l e. M ) -> ( l I l ) = .1. )
72	71	anidms 655	. . . . . . 7 |- ( l e. M -> ( l I l ) = .1. )
73	72	ad2antrl 739	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l I l ) = .1. )
74	73	oveq1d 6335	. . . . 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 6472	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l X k ) e. B )
76	2, 3, 10	ringlidm 17859	. . . . . 6 |- ( ( R e. Ring /\ ( l X k ) e. B ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
77	5, 75, 76	syl2anc 671	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
78	61, 74, 77	3eqtrd 2500	. . . 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 2500	. . 3 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l ( I F X ) k ) = ( l X k ) )
80	79	ralrimivva 2821	. 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 19481	. . . . 5 |- ( ph -> ( I F X ) e. ( B ^m ( M X. N ) ) )
82		elmapi 7524	. . . . 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 5755	. . . 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 5755	. . . 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 6435	. . 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 671	. 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 240	1 |- ( ph -> ( I F X ) = X )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   A.wral 2749   _Vcvv 3057   ifcif 3893   <.cotp 3988   |-> cmpt 4477   X. cxp 4854   Fn wfn 5600   -->wf 5601   ` cfv 5605  (class class class)co 6320   |-> cmpt2 6322   ^m cmap 7503   Fincfn 7600   Basecbs 15176   .rcmulr 15246   0gc0g 15393   gsum_g cgsu 15394   Mndcmnd 16590   1rcur 17790   Ringcrg 17835   maMul cmmul 19463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-ot 3989  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-supp 6947  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-fsupp 7915  df-oi 8056  df-card 8404  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-fzo 11953  df-seq 12252  df-hash 12554  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-0g 15395  df-gsum 15396  df-mre 15547  df-mrc 15548  df-acs 15550  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-submnd 16638  df-grp 16728  df-minusg 16729  df-mulg 16731  df-cntz 17026  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-mamu 19464

This theorem is referenced by:  matring  19523  mat1  19527