Theorem mamulid 20247

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 2622	. . . . 5 |- ( .r ` R ) = ( .r ` R )
4		mamumat1cl.r	. . . . . 6 |- ( ph -> R e. Ring )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> R e. Ring )
6		mamumat1cl.m	. . . . . 6 |- ( ph -> M e. Fin )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> M e. Fin )
8		mamulid.n	. . . . . 6 |- ( ph -> N e. Fin )
9	8	adantr 481	. . . . 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 20245	. . . . . 6 |- ( ph -> I e. ( B ^m ( M X. M ) ) )
14	13	adantr 481	. . . . 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 481	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> X e. ( B ^m ( M X. N ) ) )
17		simprl 794	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> l e. M )
18		simprr 796	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> k e. N )
19	1, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18	mamufv 20193	. . . 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 18556	. . . . . 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 762	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> R e. Ring )
23		elmapi 7879	. . . . . . . . . 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 762	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> I : ( M X. M ) --> B )
26		simplrl 800	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> l e. M )
27		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> m e. M )
28	25, 26, 27	fovrnd 6806	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( l I m ) e. B )
29		elmapi 7879	. . . . . . . . . 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 762	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> X : ( M X. N ) --> B )
32		simplrr 801	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> k e. N )
33	31, 27, 32	fovrnd 6806	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( m X k ) e. B )
34	2, 3	ringcl 18561	. . . . . . 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 1326	. . . . . 6 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) e. B )
36		eqid 2622	. . . . . 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 6385	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) : M --> B )
38	26	3adant3 1081	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> l e. M )
39		simp2 1062	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> m e. M )
40	2, 4, 10, 11, 12, 6	mat1comp 20246	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( l = m , .1. , .0. ) )
41		equcom 1945	. . . . . . . . . . . . 13 |- ( l = m <-> m = l )
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( l e. M /\ m e. M ) -> ( l = m <-> m = l ) )
43	42	ifbid 4108	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> if ( l = m , .1. , .0. ) = if ( m = l , .1. , .0. ) )
44	40, 43	eqtrd 2656	. . . . . . . . . 10 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
45	38, 39, 44	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
46		ifnefalse 4098	. . . . . . . . . 10 |- ( m =/= l -> if ( m = l , .1. , .0. ) = .0. )
47	46	3ad2ant3 1084	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> if ( m = l , .1. , .0. ) = .0. )
48	45, 47	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = .0. )
49	48	oveq1d 6665	. . . . . . 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 18587	. . . . . . . . 9 |- ( ( R e. Ring /\ ( m X k ) e. B ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
51	22, 33, 50	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
52	51	3adant3 1081	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
53	49, 52	eqtrd 2656	. . . . . 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 7330	. . . . 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 18361	. . . 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 6658	. . . . . . . 8 |- ( m = l -> ( l I m ) = ( l I l ) )
57		oveq1 6657	. . . . . . . 8 |- ( m = l -> ( m X k ) = ( l X k ) )
58	56, 57	oveq12d 6668	. . . . . . 7 |- ( m = l -> ( ( l I m ) ( .r ` R ) ( m X k ) ) = ( ( l I l ) ( .r ` R ) ( l X k ) ) )
59		ovex 6678	. . . . . . 7 |- ( ( l I l ) ( .r ` R ) ( l X k ) ) e. _V
60	58, 36, 59	fvmpt 6282	. . . . . 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 764	. . . . 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 1952	. . . . . . . . . 10 |- ( i = l -> ( i = j <-> l = j ) )
63	62	ifbid 4108	. . . . . . . . 9 |- ( i = l -> if ( i = j , .1. , .0. ) = if ( l = j , .1. , .0. ) )
64		equequ2 1953	. . . . . . . . . . 11 |- ( j = l -> ( l = j <-> l = l ) )
65	64	ifbid 4108	. . . . . . . . . 10 |- ( j = l -> if ( l = j , .1. , .0. ) = if ( l = l , .1. , .0. ) )
66		equid 1939	. . . . . . . . . . 11 |- l = l
67	66	iftruei 4093	. . . . . . . . . 10 |- if ( l = l , .1. , .0. ) = .1.
68	65, 67	syl6eq 2672	. . . . . . . . 9 |- ( j = l -> if ( l = j , .1. , .0. ) = .1. )
69		fvex 6201	. . . . . . . . . 10 |- ( 1r ` R ) e. _V
70	10, 69	eqeltri 2697	. . . . . . . . 9 |- .1. e. _V
71	63, 68, 12, 70	ovmpt2 6796	. . . . . . . 8 |- ( ( l e. M /\ l e. M ) -> ( l I l ) = .1. )
72	71	anidms 677	. . . . . . 7 |- ( l e. M -> ( l I l ) = .1. )
73	72	ad2antrl 764	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l I l ) = .1. )
74	73	oveq1d 6665	. . . . 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 6805	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l X k ) e. B )
76	2, 3, 10	ringlidm 18571	. . . . . 6 |- ( ( R e. Ring /\ ( l X k ) e. B ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
77	5, 75, 76	syl2anc 693	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
78	61, 74, 77	3eqtrd 2660	. . . 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 2660	. . 3 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l ( I F X ) k ) = ( l X k ) )
80	79	ralrimivva 2971	. 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 20207	. . . . 5 |- ( ph -> ( I F X ) e. ( B ^m ( M X. N ) ) )
82		elmapi 7879	. . . . 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 6045	. . . 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 6045	. . . 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 6767	. . 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 693	. 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 247	1 |- ( ph -> ( I F X ) = X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   ifcif 4086   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   Basecbs 15857   .rcmulr 15942   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   1rcur 18501   Ringcrg 18547   maMul cmmul 20189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-mamu 20190

This theorem is referenced by:  matring  20249  mat1  20253