Theorem mamurid 20248

Description: The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (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 )
mamurid.f	|- F = ( R maMul <. N , M , M >. )
mamurid.x	|- ( ph -> X e. ( B ^m ( N X. M ) ) )

Assertion

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

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

Step	Hyp	Ref	Expression
1		mamurid.f	. . . . 5 |- F = ( R maMul <. N , M , M >. )
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. N /\ m e. M ) ) -> R e. Ring )
6		mamulid.n	. . . . . 6 |- ( ph -> N e. Fin )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> N e. Fin )
8		mamumat1cl.m	. . . . . 6 |- ( ph -> M e. Fin )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> M e. Fin )
10		mamurid.x	. . . . . 6 |- ( ph -> X e. ( B ^m ( N X. M ) ) )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> X e. ( B ^m ( N X. M ) ) )
12		mamumat1cl.o	. . . . . . 7 |- .1. = ( 1r ` R )
13		mamumat1cl.z	. . . . . . 7 |- .0. = ( 0g ` R )
14		mamumat1cl.i	. . . . . . 7 |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
15	2, 4, 12, 13, 14, 8	mamumat1cl 20245	. . . . . 6 |- ( ph -> I e. ( B ^m ( M X. M ) ) )
16	15	adantr 481	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> I e. ( B ^m ( M X. M ) ) )
17		simprl 794	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> l e. N )
18		simprr 796	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> m e. M )
19	1, 2, 3, 5, 7, 9, 9, 11, 16, 17, 18	mamufv 20193	. . . 4 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( l ( X F I ) m ) = ( R gsumg ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) ) )
20		ringmnd 18556	. . . . . 6 |- ( R e. Ring -> R e. Mnd )
21	5, 20	syl 17	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> R e. Mnd )
22	4	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> R e. Ring )
23		elmapi 7879	. . . . . . . . . 10 |- ( X e. ( B ^m ( N X. M ) ) -> X : ( N X. M ) --> B )
24	10, 23	syl 17	. . . . . . . . 9 |- ( ph -> X : ( N X. M ) --> B )
25	24	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> X : ( N X. M ) --> B )
26		simplrl 800	. . . . . . . 8 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> l e. N )
27		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> k e. M )
28	25, 26, 27	fovrnd 6806	. . . . . . 7 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> ( l X k ) e. B )
29		elmapi 7879	. . . . . . . . . 10 |- ( I e. ( B ^m ( M X. M ) ) -> I : ( M X. M ) --> B )
30	15, 29	syl 17	. . . . . . . . 9 |- ( ph -> I : ( M X. M ) --> B )
31	30	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> I : ( M X. M ) --> B )
32		simplrr 801	. . . . . . . 8 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> m e. M )
33	31, 27, 32	fovrnd 6806	. . . . . . 7 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> ( k I m ) e. B )
34	2, 3	ringcl 18561	. . . . . . 7 |- ( ( R e. Ring /\ ( l X k ) e. B /\ ( k I m ) e. B ) -> ( ( l X k ) ( .r ` R ) ( k I m ) ) e. B )
35	22, 28, 33, 34	syl3anc 1326	. . . . . 6 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> ( ( l X k ) ( .r ` R ) ( k I m ) ) e. B )
36		eqid 2622	. . . . . 6 |- ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) = ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) )
37	35, 36	fmptd 6385	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) : M --> B )
38		simp2 1062	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> k e. M )
39	32	3adant3 1081	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> m e. M )
40	2, 4, 12, 13, 14, 8	mat1comp 20246	. . . . . . . . . 10 |- ( ( k e. M /\ m e. M ) -> ( k I m ) = if ( k = m , .1. , .0. ) )
41	38, 39, 40	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> ( k I m ) = if ( k = m , .1. , .0. ) )
42		ifnefalse 4098	. . . . . . . . . 10 |- ( k =/= m -> if ( k = m , .1. , .0. ) = .0. )
43	42	3ad2ant3 1084	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> if ( k = m , .1. , .0. ) = .0. )
44	41, 43	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> ( k I m ) = .0. )
45	44	oveq2d 6666	. . . . . . 7 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> ( ( l X k ) ( .r ` R ) ( k I m ) ) = ( ( l X k ) ( .r ` R ) .0. ) )
46	2, 3, 13	ringrz 18588	. . . . . . . . 9 |- ( ( R e. Ring /\ ( l X k ) e. B ) -> ( ( l X k ) ( .r ` R ) .0. ) = .0. )
47	22, 28, 46	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M ) -> ( ( l X k ) ( .r ` R ) .0. ) = .0. )
48	47	3adant3 1081	. . . . . . 7 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> ( ( l X k ) ( .r ` R ) .0. ) = .0. )
49	45, 48	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ ( l e. N /\ m e. M ) ) /\ k e. M /\ k =/= m ) -> ( ( l X k ) ( .r ` R ) ( k I m ) ) = .0. )
50	49, 9	suppsssn 7330	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) supp .0. ) C_ { m } )
51	2, 13, 21, 9, 18, 37, 50	gsumpt 18361	. . . 4 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( R gsumg ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) ) = ( ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) ` m ) )
52		oveq2 6658	. . . . . . . 8 |- ( k = m -> ( l X k ) = ( l X m ) )
53		oveq1 6657	. . . . . . . 8 |- ( k = m -> ( k I m ) = ( m I m ) )
54	52, 53	oveq12d 6668	. . . . . . 7 |- ( k = m -> ( ( l X k ) ( .r ` R ) ( k I m ) ) = ( ( l X m ) ( .r ` R ) ( m I m ) ) )
55		ovex 6678	. . . . . . 7 |- ( ( l X m ) ( .r ` R ) ( m I m ) ) e. _V
56	54, 36, 55	fvmpt 6282	. . . . . 6 |- ( m e. M -> ( ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) ` m ) = ( ( l X m ) ( .r ` R ) ( m I m ) ) )
57	56	ad2antll 765	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) ` m ) = ( ( l X m ) ( .r ` R ) ( m I m ) ) )
58		equequ1 1952	. . . . . . . . . 10 |- ( i = m -> ( i = j <-> m = j ) )
59	58	ifbid 4108	. . . . . . . . 9 |- ( i = m -> if ( i = j , .1. , .0. ) = if ( m = j , .1. , .0. ) )
60		equequ2 1953	. . . . . . . . . . 11 |- ( j = m -> ( m = j <-> m = m ) )
61	60	ifbid 4108	. . . . . . . . . 10 |- ( j = m -> if ( m = j , .1. , .0. ) = if ( m = m , .1. , .0. ) )
62		eqid 2622	. . . . . . . . . . 11 |- m = m
63	62	iftruei 4093	. . . . . . . . . 10 |- if ( m = m , .1. , .0. ) = .1.
64	61, 63	syl6eq 2672	. . . . . . . . 9 |- ( j = m -> if ( m = j , .1. , .0. ) = .1. )
65		fvex 6201	. . . . . . . . . 10 |- ( 1r ` R ) e. _V
66	12, 65	eqeltri 2697	. . . . . . . . 9 |- .1. e. _V
67	59, 64, 14, 66	ovmpt2 6796	. . . . . . . 8 |- ( ( m e. M /\ m e. M ) -> ( m I m ) = .1. )
68	67	anidms 677	. . . . . . 7 |- ( m e. M -> ( m I m ) = .1. )
69	68	oveq2d 6666	. . . . . 6 |- ( m e. M -> ( ( l X m ) ( .r ` R ) ( m I m ) ) = ( ( l X m ) ( .r ` R ) .1. ) )
70	69	ad2antll 765	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( ( l X m ) ( .r ` R ) ( m I m ) ) = ( ( l X m ) ( .r ` R ) .1. ) )
71	24	fovrnda 6805	. . . . . 6 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( l X m ) e. B )
72	2, 3, 12	ringridm 18572	. . . . . 6 |- ( ( R e. Ring /\ ( l X m ) e. B ) -> ( ( l X m ) ( .r ` R ) .1. ) = ( l X m ) )
73	5, 71, 72	syl2anc 693	. . . . 5 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( ( l X m ) ( .r ` R ) .1. ) = ( l X m ) )
74	57, 70, 73	3eqtrd 2660	. . . 4 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( ( k e. M |-> ( ( l X k ) ( .r ` R ) ( k I m ) ) ) ` m ) = ( l X m ) )
75	19, 51, 74	3eqtrd 2660	. . 3 |- ( ( ph /\ ( l e. N /\ m e. M ) ) -> ( l ( X F I ) m ) = ( l X m ) )
76	75	ralrimivva 2971	. 2 |- ( ph -> A. l e. N A. m e. M ( l ( X F I ) m ) = ( l X m ) )
77	2, 4, 1, 6, 8, 8, 10, 15	mamucl 20207	. . . . 5 |- ( ph -> ( X F I ) e. ( B ^m ( N X. M ) ) )
78		elmapi 7879	. . . . 5 |- ( ( X F I ) e. ( B ^m ( N X. M ) ) -> ( X F I ) : ( N X. M ) --> B )
79	77, 78	syl 17	. . . 4 |- ( ph -> ( X F I ) : ( N X. M ) --> B )
80		ffn 6045	. . . 4 |- ( ( X F I ) : ( N X. M ) --> B -> ( X F I ) Fn ( N X. M ) )
81	79, 80	syl 17	. . 3 |- ( ph -> ( X F I ) Fn ( N X. M ) )
82		ffn 6045	. . . 4 |- ( X : ( N X. M ) --> B -> X Fn ( N X. M ) )
83	24, 82	syl 17	. . 3 |- ( ph -> X Fn ( N X. M ) )
84		eqfnov2 6767	. . 3 |- ( ( ( X F I ) Fn ( N X. M ) /\ X Fn ( N X. M ) ) -> ( ( X F I ) = X <-> A. l e. N A. m e. M ( l ( X F I ) m ) = ( l X m ) ) )
85	81, 83, 84	syl2anc 693	. 2 |- ( ph -> ( ( X F I ) = X <-> A. l e. N A. m e. M ( l ( X F I ) m ) = ( l X m ) ) )
86	76, 85	mpbird 247	1 |- ( ph -> ( X F I ) = 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