Theorem mamulid 18413

Description: Diagonal matrices are left identities. (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)

Hypotheses

Ref	Expression
mamucl.b	|- B = ( Base ` R )
mamucl.r	|- ( ph -> R e. Ring )
mamudiag.o	|- .1. = ( 1r ` R )
mamudiag.z	|- .0. = ( 0g ` R )
mamudiag.i	|- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
mamudiag.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   .1. , i, j   .0. , 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 l m k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mamulid.f	. . . . 5 |- F = ( R maMul <. M , M , N >. )
2		mamucl.b	. . . . 5 |- B = ( Base ` R )
3		eqid 2451	. . . . 5 |- ( .r ` R ) = ( .r ` R )
4		mamucl.r	. . . . . 6 |- ( ph -> R e. Ring )
5	4	adantr 465	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> R e. Ring )
6		mamudiag.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		mamudiag.o	. . . . . . 7 |- .1. = ( 1r ` R )
11		mamudiag.z	. . . . . . 7 |- .0. = ( 0g ` R )
12		mamudiag.i	. . . . . . 7 |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
13	2, 4, 10, 11, 12, 6	mamudiagcl 18411	. . . . . 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 18394	. . . 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 16760	. . . . . 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 7334	. . . . . . . . . 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 6335	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( l I m ) e. B )
29		elmapi 7334	. . . . . . . . . 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 6335	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( m X k ) e. B )
34	2, 3	rngcl 16764	. . . . . . 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 1219	. . . . . 6 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M ) -> ( ( l I m ) ( .r ` R ) ( m X k ) ) e. B )
36		eqid 2451	. . . . . 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 5966	. . . . 5 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( m e. M |-> ( ( l I m ) ( .r ` R ) ( m X k ) ) ) : M --> B )
38	26	3adant3 1008	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> l e. M )
39		simp2 989	. . . . . . . . . 10 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> m e. M )
40	2, 4, 10, 11, 12, 6	dmatcomp 18412	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> ( l I m ) = if ( l = m , .1. , .0. ) )
41		equcom 1734	. . . . . . . . . . . . 13 |- ( l = m <-> m = l )
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( l e. M /\ m e. M ) -> ( l = m <-> m = l ) )
43	42	ifbid 3909	. . . . . . . . . . 11 |- ( ( l e. M /\ m e. M ) -> if ( l = m , .1. , .0. ) = if ( m = l , .1. , .0. ) )
44	40, 43	eqtrd 2492	. . . . . . . . . 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 3899	. . . . . . . . . 10 |- ( m =/= l -> if ( m = l , .1. , .0. ) = .0. )
47	46	3ad2ant3 1011	. . . . . . . . 9 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> if ( m = l , .1. , .0. ) = .0. )
48	45, 47	eqtrd 2492	. . . . . . . 8 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( l I m ) = .0. )
49	48	oveq1d 6205	. . . . . . 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 16787	. . . . . . . . 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 1008	. . . . . . 7 |- ( ( ( ph /\ ( l e. M /\ k e. N ) ) /\ m e. M /\ m =/= l ) -> ( .0. ( .r ` R ) ( m X k ) ) = .0. )
53	49, 52	eqtrd 2492	. . . . . 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 6824	. . . . 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 16559	. . . 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 6198	. . . . . . . 8 |- ( m = l -> ( l I m ) = ( l I l ) )
57		oveq1 6197	. . . . . . . 8 |- ( m = l -> ( m X k ) = ( l X k ) )
58	56, 57	oveq12d 6208	. . . . . . 7 |- ( m = l -> ( ( l I m ) ( .r ` R ) ( m X k ) ) = ( ( l I l ) ( .r ` R ) ( l X k ) ) )
59		ovex 6215	. . . . . . 7 |- ( ( l I l ) ( .r ` R ) ( l X k ) ) e. _V
60	58, 36, 59	fvmpt 5873	. . . . . 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 1738	. . . . . . . . . 10 |- ( i = l -> ( i = j <-> l = j ) )
63	62	ifbid 3909	. . . . . . . . 9 |- ( i = l -> if ( i = j , .1. , .0. ) = if ( l = j , .1. , .0. ) )
64		equequ2 1739	. . . . . . . . . . 11 |- ( j = l -> ( l = j <-> l = l ) )
65	64	ifbid 3909	. . . . . . . . . 10 |- ( j = l -> if ( l = j , .1. , .0. ) = if ( l = l , .1. , .0. ) )
66		equid 1731	. . . . . . . . . . 11 |- l = l
67	66	iftruei 3896	. . . . . . . . . 10 |- if ( l = l , .1. , .0. ) = .1.
68	65, 67	syl6eq 2508	. . . . . . . . 9 |- ( j = l -> if ( l = j , .1. , .0. ) = .1. )
69		fvex 5799	. . . . . . . . . 10 |- ( 1r ` R ) e. _V
70	10, 69	eqeltri 2535	. . . . . . . . 9 |- .1. e. _V
71	63, 68, 12, 70	ovmpt2 6326	. . . . . . . 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 6205	. . . . 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 6334	. . . . . 6 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l X k ) e. B )
76	2, 3, 10	rnglidm 16774	. . . . . 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 2496	. . . 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 2496	. . 3 |- ( ( ph /\ ( l e. M /\ k e. N ) ) -> ( l ( I F X ) k ) = ( l X k ) )
80	79	ralrimivva 2904	. 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 18410	. . . . 5 |- ( ph -> ( I F X ) e. ( B ^m ( M X. N ) ) )
82		elmapi 7334	. . . . 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 5657	. . . 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 5657	. . . 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 6297	. . 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 965   = wceq 1370   e. wcel 1758   =/= wne 2644   A.wral 2795   _Vcvv 3068   ifcif 3889   <.cotp 3983   |-> cmpt 4448   X. cxp 4936   Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   |-> cmpt2 6192   ^m cmap 7314   Fincfn 7410   Basecbs 14276   .rcmulr 14341   0gc0g 14480   gsum_g cgsu 14481   Mndcmnd 15511   1rcur 16708   Ringcrg 16751   maMul cmmul 18388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-ot 3984  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-0g 14482  df-gsum 14483  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-grp 15647  df-minusg 15648  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-mamu 18390

This theorem is referenced by:  matrng  18440  mat1  18445