Theorem mamulid 18710

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 2467	. . . . 5 |- ( .r ` R ) = ( .r ` R )
4		mamumat1cl.r	. . . . . 6 |- ( ph -> R e. Ring )
5	4	adantr 465	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> R e. Ring )
6		mamumat1cl.m	. . . . . 6 |- ( ph -> M e. Fin )
7	6	adantr 465	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> M e. Fin )
8		mamulid.n	. . . . . 6 |- ( ph -> N e. Fin )
9	8	adantr 465	. . . . 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 18708	. . . . . 6 |- ( ph -> I e. ( B ^m ( M X. M ) ) )
14	13	adantr 465	. . . . 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 465	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> X e. ( B ^m ( M X. N ) ) )
17		simprl 755	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> l e. M )
18		simprr 756	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> k e. N )
19	1, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18	mamufv 18656	. . . 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		rngmnd 16995	. . . . . 6 |- ( R e. Ring -> R e. Mnd )
21	5, 20	syl 16	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> R e. Mnd )
22	4	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> R e. Ring )
23		elmapi 7437	. . . . . . . . . 10 |- ( I e. ( B ^m ( M X. M ) ) -> I : ( M X. M ) --> B )
24	13, 23	syl 16	. . . . . . . . 9 |- ( ph -> I : ( M X. M ) --> B )
25	24	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> I : ( M X. M ) --> B )
26		simplrl 759	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> l e. M )
27		simpr 461	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> m e. M )
28	25, 26, 27	fovrnd 6429	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( l I m ) e. B )
29		elmapi 7437	. . . . . . . . . 10 |- ( X e. ( B ^m ( M X. N ) ) -> X : ( M X. N ) --> B )
30	15, 29	syl 16	. . . . . . . . 9 |- ( ph -> X : ( M X. N ) --> B )
31	30	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> X : ( M X. N ) --> B )
32		simplrr 760	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> k e. N )
33	31, 27, 32	fovrnd 6429	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( m X k ) e. B )
34	2, 3	rngcl 16999	. . . . . . 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 1228	. . . . . 6 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) e. B )
36		eqid 2467	. . . . . 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 6043	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) : M --> B )
38	26	3adant3 1016	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> l e. M )
39		simp2 997	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> m e. M )
40	2, 4, 10, 11, 12, 6	mat1comp 18709	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( l = m , .1. , .0. ) )
41		equcom 1743	. . . . . . . . . . . . 13 |- ( l = m <-> m = l )
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( l e. M /\ m e. M ) -> ( l = m <-> m = l ) )
43	42	ifbid 3961	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> if ( l = m , .1. , .0. ) = if ( m = l , .1. , .0. ) )
44	40, 43	eqtrd 2508	. . . . . . . . . 10 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
45	38, 39, 44	syl2anc 661	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = if ( m = l , .1. , .0. ) )
46		ifnefalse 3951	. . . . . . . . . 10 |- ( m =/= l -> if ( m = l , .1. , .0. ) = .0. )
47	46	3ad2ant3 1019	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> if ( m = l , .1. , .0. ) = .0. )
48	45, 47	eqtrd 2508	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = .0. )
49	48	oveq1d 6297	. . . . . . 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	rnglz 17022	. . . . . . . . 9 |- ( ( R e. Ring /\ ( m X k ) e. B ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
51	22, 33, 50	syl2anc 661	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
52	51	3adant3 1016	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
53	49, 52	eqtrd 2508	. . . . . 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 6932	. . . . 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 16779	. . . 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 6290	. . . . . . . 8 |- ( m = l -> ( l I m ) = ( l I l ) )
57		oveq1 6289	. . . . . . . 8 |- ( m = l -> ( m X k ) = ( l X k ) )
58	56, 57	oveq12d 6300	. . . . . . 7 |- ( m = l -> ( ( l I m ) ( .r ` R ) ( m X k ) ) = ( ( l I l ) ( .r ` R ) ( l X k ) ) )
59		ovex 6307	. . . . . . 7 |- ( ( l I l ) ( .r ` R ) ( l X k ) ) e. _V
60	58, 36, 59	fvmpt 5948	. . . . . 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 727	. . . . 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 1747	. . . . . . . . . 10 |- ( i = l -> ( i = j <-> l = j ) )
63	62	ifbid 3961	. . . . . . . . 9 |- ( i = l -> if ( i = j , .1. , .0. ) = if ( l = j , .1. , .0. ) )
64		equequ2 1748	. . . . . . . . . . 11 |- ( j = l -> ( l = j <-> l = l ) )
65	64	ifbid 3961	. . . . . . . . . 10 |- ( j = l -> if ( l = j , .1. , .0. ) = if ( l = l , .1. , .0. ) )
66		equid 1740	. . . . . . . . . . 11 |- l = l
67	66	iftruei 3946	. . . . . . . . . 10 |- if ( l = l , .1. , .0. ) = .1.
68	65, 67	syl6eq 2524	. . . . . . . . 9 |- ( j = l -> if ( l = j , .1. , .0. ) = .1. )
69		fvex 5874	. . . . . . . . . 10 |- ( 1r ` R ) e. _V
70	10, 69	eqeltri 2551	. . . . . . . . 9 |- .1. e. _V
71	63, 68, 12, 70	ovmpt2 6420	. . . . . . . 8 |- ( ( l e. M /\ l e. M ) -> ( l I l ) = .1. )
72	71	anidms 645	. . . . . . 7 |- ( l e. M -> ( l I l ) = .1. )
73	72	ad2antrl 727	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l I l ) = .1. )
74	73	oveq1d 6297	. . . . 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 6428	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l X k ) e. B )
76	2, 3, 10	rnglidm 17009	. . . . . 6 |- ( ( R e. Ring /\ ( l X k ) e. B ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
77	5, 75, 76	syl2anc 661	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( .1. ( .r ` R ) ( l X k ) ) = ( l X k ) )
78	61, 74, 77	3eqtrd 2512	. . . 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 2512	. . 3 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l ( I F X ) k ) = ( l X k ) )
80	79	ralrimivva 2885	. 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 18670	. . . . 5 |- ( ph -> ( I F X ) e. ( B ^m ( M X. N ) ) )
82		elmapi 7437	. . . . 5 |- ( ( I F X ) e. ( B ^m ( M X. N ) ) -> ( I F X ) : ( M X. N ) --> B )
83	81, 82	syl 16	. . . 4 |- ( ph -> ( I F X ) : ( M X. N ) --> B )
84		ffn 5729	. . . 4 |- ( ( I F X ) : ( M X. N ) --> B -> ( I F X ) Fn ( M X. N ) )
85	83, 84	syl 16	. . 3 |- ( ph -> ( I F X ) Fn ( M X. N ) )
86		ffn 5729	. . . 4 |- ( X : ( M X. N ) --> B -> X Fn ( M X. N ) )
87	30, 86	syl 16	. . 3 |- ( ph -> X Fn ( M X. N ) )
88		eqfnov2 6391	. . 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 661	. 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 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   _Vcvv 3113   ifcif 3939   <.cotp 4035   |-> cmpt 4505   X. cxp 4997   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   ^m cmap 7417   Fincfn 7513   Basecbs 14486   .rcmulr 14552   0gc0g 14691   gsum_g cgsu 14692   Mndcmnd 15722   1rcur 16943   Ringcrg 16986   maMul cmmul 18652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-gsum 14694  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-mamu 18653

This theorem is referenced by:  matrng  18712  mat1  18716