Theorem madurid 19600

Description: Multiplying a matrix with its adjunct results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 16-Jul-2018.)

Hypotheses

Ref	Expression
madurid.a	|- A = ( N Mat R )
madurid.b	|- B = ( Base ` A )
madurid.j	|- J = ( N maAdju R )
madurid.d	|- D = ( N maDet R )
madurid.i	|- .1. = ( 1r ` A )
madurid.t	|- .x. = ( .r ` A )
madurid.s	|- .xb = ( .s ` A )

Assertion

Ref	Expression
madurid	|- ( ( M e. B /\ R e. CRing ) -> ( M .x. ( J ` M ) ) = ( ( D ` M ) .xb .1. ) )

Proof of Theorem madurid

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2429	. . 3 |- ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
2		eqid 2429	. . 3 |- ( Base ` R ) = ( Base ` R )
3		eqid 2429	. . 3 |- ( .r ` R ) = ( .r ` R )
4		simpr 462	. . 3 |- ( ( M e. B /\ R e. CRing ) -> R e. CRing )
5		madurid.a	. . . . . 6 |- A = ( N Mat R )
6		madurid.b	. . . . . 6 |- B = ( Base ` A )
7	5, 6	matrcl 19368	. . . . 5 |- ( M e. B -> ( N e. Fin /\ R e. _V ) )
8	7	simpld 460	. . . 4 |- ( M e. B -> N e. Fin )
9	8	adantr 466	. . 3 |- ( ( M e. B /\ R e. CRing ) -> N e. Fin )
10	5, 2, 6	matbas2i 19378	. . . 4 |- ( M e. B -> M e. ( ( Base ` R ) ^m ( N X. N ) ) )
11	10	adantr 466	. . 3 |- ( ( M e. B /\ R e. CRing ) -> M e. ( ( Base ` R ) ^m ( N X. N ) ) )
12		madurid.j	. . . . . . 7 |- J = ( N maAdju R )
13	5, 12, 6	maduf 19597	. . . . . 6 |- ( R e. CRing -> J : B --> B )
14	13	adantl 467	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> J : B --> B )
15		simpl 458	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> M e. B )
16	14, 15	ffvelrnd 6038	. . . 4 |- ( ( M e. B /\ R e. CRing ) -> ( J ` M ) e. B )
17	5, 2, 6	matbas2i 19378	. . . 4 |- ( ( J ` M ) e. B -> ( J ` M ) e. ( ( Base ` R ) ^m ( N X. N ) ) )
18	16, 17	syl 17	. . 3 |- ( ( M e. B /\ R e. CRing ) -> ( J ` M ) e. ( ( Base ` R ) ^m ( N X. N ) ) )
19	1, 2, 3, 4, 9, 9, 9, 11, 18	mamuval 19342	. 2 |- ( ( M e. B /\ R e. CRing ) -> ( M ( R maMul <. N , N , N >. ) ( J ` M ) ) = ( a e. N , b e. N |-> ( R gsumg ( c e. N |-> ( ( a M c ) ( .r ` R ) ( c ( J ` M ) b ) ) ) ) ) )
20	5, 1	matmulr 19394	. . . . 5 |- ( ( N e. Fin /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = ( .r ` A ) )
21	8, 20	sylan 473	. . . 4 |- ( ( M e. B /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = ( .r ` A ) )
22		madurid.t	. . . 4 |- .x. = ( .r ` A )
23	21, 22	syl6eqr 2488	. . 3 |- ( ( M e. B /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = .x. )
24	23	oveqd 6322	. 2 |- ( ( M e. B /\ R e. CRing ) -> ( M ( R maMul <. N , N , N >. ) ( J ` M ) ) = ( M .x. ( J ` M ) ) )
25		madurid.d	. . . . . 6 |- D = ( N maDet R )
26		simp1l 1029	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> M e. B )
27		simp1r 1030	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> R e. CRing )
28		elmapi 7501	. . . . . . . . . 10 |- ( M e. ( ( Base ` R ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` R ) )
29	11, 28	syl 17	. . . . . . . . 9 |- ( ( M e. B /\ R e. CRing ) -> M : ( N X. N ) --> ( Base ` R ) )
30	29	3ad2ant1 1026	. . . . . . . 8 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
31	30	adantr 466	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
32		simpl2 1009	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> a e. N )
33		simpr 462	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> c e. N )
34	31, 32, 33	fovrnd 6455	. . . . . 6 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> ( a M c ) e. ( Base ` R ) )
35		simp3 1007	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> b e. N )
36	5, 12, 6, 25, 3, 2, 26, 27, 34, 35	madugsum 19599	. . . . 5 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> ( R gsumg ( c e. N |-> ( ( a M c ) ( .r ` R ) ( c ( J ` M ) b ) ) ) ) = ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) ) )
37		iftrue 3921	. . . . . . . . 9 |- ( a = b -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( D ` M ) )
38	37	adantl 467	. . . . . . . 8 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( D ` M ) )
39		ffn 5746	. . . . . . . . . . . . 13 |- ( M : ( N X. N ) --> ( Base ` R ) -> M Fn ( N X. N ) )
40	29, 39	syl 17	. . . . . . . . . . . 12 |- ( ( M e. B /\ R e. CRing ) -> M Fn ( N X. N ) )
41		fnov 6418	. . . . . . . . . . . 12 |- ( M Fn ( N X. N ) <-> M = ( d e. N , c e. N |-> ( d M c ) ) )
42	40, 41	sylib 199	. . . . . . . . . . 11 |- ( ( M e. B /\ R e. CRing ) -> M = ( d e. N , c e. N |-> ( d M c ) ) )
43	42	adantr 466	. . . . . . . . . 10 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> M = ( d e. N , c e. N |-> ( d M c ) ) )
44		equtr2 1854	. . . . . . . . . . . . . . 15 |- ( ( a = b /\ d = b ) -> a = d )
45	44	oveq1d 6320	. . . . . . . . . . . . . 14 |- ( ( a = b /\ d = b ) -> ( a M c ) = ( d M c ) )
46	45	ifeq1da 3945	. . . . . . . . . . . . 13 |- ( a = b -> if ( d = b , ( a M c ) , ( d M c ) ) = if ( d = b , ( d M c ) , ( d M c ) ) )
47		ifid 3952	. . . . . . . . . . . . 13 |- if ( d = b , ( d M c ) , ( d M c ) ) = ( d M c )
48	46, 47	syl6eq 2486	. . . . . . . . . . . 12 |- ( a = b -> if ( d = b , ( a M c ) , ( d M c ) ) = ( d M c ) )
49	48	adantl 467	. . . . . . . . . . 11 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> if ( d = b , ( a M c ) , ( d M c ) ) = ( d M c ) )
50	49	mpt2eq3dv 6371	. . . . . . . . . 10 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) = ( d e. N , c e. N |-> ( d M c ) ) )
51	43, 50	eqtr4d 2473	. . . . . . . . 9 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> M = ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) )
52	51	fveq2d 5885	. . . . . . . 8 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> ( D ` M ) = ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) ) )
53	38, 52	eqtr2d 2471	. . . . . . 7 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) ) = if ( a = b , ( D ` M ) , ( 0g ` R ) ) )
54	53	3ad2antl1 1167	. . . . . 6 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ a = b ) -> ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) ) = if ( a = b , ( D ` M ) , ( 0g ` R ) ) )
55		eqid 2429	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
56		simpl1r 1057	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> R e. CRing )
57	9	3ad2ant1 1026	. . . . . . . . 9 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> N e. Fin )
58	57	adantr 466	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> N e. Fin )
59	30	ad2antrr 730	. . . . . . . . 9 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ c e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
60		simpll2 1045	. . . . . . . . 9 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ c e. N ) -> a e. N )
61		simpr 462	. . . . . . . . 9 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ c e. N ) -> c e. N )
62	59, 60, 61	fovrnd 6455	. . . . . . . 8 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ c e. N ) -> ( a M c ) e. ( Base ` R ) )
63	30	adantr 466	. . . . . . . . . 10 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> M : ( N X. N ) --> ( Base ` R ) )
64	63	fovrnda 6454	. . . . . . . . 9 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ ( d e. N /\ c e. N ) ) -> ( d M c ) e. ( Base ` R ) )
65	64	3impb 1201	. . . . . . . 8 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ d e. N /\ c e. N ) -> ( d M c ) e. ( Base ` R ) )
66		simpl3 1010	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> b e. N )
67		simpl2 1009	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> a e. N )
68		df-ne 2627	. . . . . . . . . . 11 |- ( a =/= b <-> -. a = b )
69	68	biimpri 209	. . . . . . . . . 10 |- ( -. a = b -> a =/= b )
70	69	necomd 2702	. . . . . . . . 9 |- ( -. a = b -> b =/= a )
71	70	adantl 467	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> b =/= a )
72	25, 2, 55, 56, 58, 62, 65, 66, 67, 71	mdetralt2 19565	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , if ( d = a , ( a M c ) , ( d M c ) ) ) ) ) = ( 0g ` R ) )
73		ifid 3952	. . . . . . . . . . 11 |- if ( d = a , ( d M c ) , ( d M c ) ) = ( d M c )
74		oveq1 6312	. . . . . . . . . . . . 13 |- ( d = a -> ( d M c ) = ( a M c ) )
75	74	adantl 467	. . . . . . . . . . . 12 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ d = a ) -> ( d M c ) = ( a M c ) )
76	75	ifeq1da 3945	. . . . . . . . . . 11 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> if ( d = a , ( d M c ) , ( d M c ) ) = if ( d = a , ( a M c ) , ( d M c ) ) )
77	73, 76	syl5eqr 2484	. . . . . . . . . 10 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> ( d M c ) = if ( d = a , ( a M c ) , ( d M c ) ) )
78	77	ifeq2d 3934	. . . . . . . . 9 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> if ( d = b , ( a M c ) , ( d M c ) ) = if ( d = b , ( a M c ) , if ( d = a , ( a M c ) , ( d M c ) ) ) )
79	78	mpt2eq3dv 6371	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) = ( d e. N , c e. N |-> if ( d = b , ( a M c ) , if ( d = a , ( a M c ) , ( d M c ) ) ) ) )
80	79	fveq2d 5885	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) ) = ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , if ( d = a , ( a M c ) , ( d M c ) ) ) ) ) )
81		iffalse 3924	. . . . . . . 8 |- ( -. a = b -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( 0g ` R ) )
82	81	adantl 467	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( 0g ` R ) )
83	72, 80, 82	3eqtr4d 2480	. . . . . 6 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) ) = if ( a = b , ( D ` M ) , ( 0g ` R ) ) )
84	54, 83	pm2.61dan 798	. . . . 5 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> ( D ` ( d e. N , c e. N |-> if ( d = b , ( a M c ) , ( d M c ) ) ) ) = if ( a = b , ( D ` M ) , ( 0g ` R ) ) )
85	36, 84	eqtrd 2470	. . . 4 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> ( R gsumg ( c e. N |-> ( ( a M c ) ( .r ` R ) ( c ( J ` M ) b ) ) ) ) = if ( a = b , ( D ` M ) , ( 0g ` R ) ) )
86	85	mpt2eq3dva 6369	. . 3 |- ( ( M e. B /\ R e. CRing ) -> ( a e. N , b e. N |-> ( R gsumg ( c e. N |-> ( ( a M c ) ( .r ` R ) ( c ( J ` M ) b ) ) ) ) ) = ( a e. N , b e. N |-> if ( a = b , ( D ` M ) , ( 0g ` R ) ) ) )
87		madurid.i	. . . . 5 |- .1. = ( 1r ` A )
88	87	oveq2i 6316	. . . 4 |- ( ( D ` M ) .xb .1. ) = ( ( D ` M ) .xb ( 1r ` A ) )
89		crngring 17726	. . . . . 6 |- ( R e. CRing -> R e. Ring )
90	89	adantl 467	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> R e. Ring )
91	25, 5, 6, 2	mdetf 19551	. . . . . . 7 |- ( R e. CRing -> D : B --> ( Base ` R ) )
92	91	adantl 467	. . . . . 6 |- ( ( M e. B /\ R e. CRing ) -> D : B --> ( Base ` R ) )
93	92, 15	ffvelrnd 6038	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> ( D ` M ) e. ( Base ` R ) )
94		madurid.s	. . . . . 6 |- .xb = ( .s ` A )
95	5, 2, 94, 55	matsc 19406	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ ( D ` M ) e. ( Base ` R ) ) -> ( ( D ` M ) .xb ( 1r ` A ) ) = ( a e. N , b e. N |-> if ( a = b , ( D ` M ) , ( 0g ` R ) ) ) )
96	9, 90, 93, 95	syl3anc 1264	. . . 4 |- ( ( M e. B /\ R e. CRing ) -> ( ( D ` M ) .xb ( 1r ` A ) ) = ( a e. N , b e. N |-> if ( a = b , ( D ` M ) , ( 0g ` R ) ) ) )
97	88, 96	syl5eq 2482	. . 3 |- ( ( M e. B /\ R e. CRing ) -> ( ( D ` M ) .xb .1. ) = ( a e. N , b e. N |-> if ( a = b , ( D ` M ) , ( 0g ` R ) ) ) )
98	86, 97	eqtr4d 2473	. 2 |- ( ( M e. B /\ R e. CRing ) -> ( a e. N , b e. N |-> ( R gsumg ( c e. N |-> ( ( a M c ) ( .r ` R ) ( c ( J ` M ) b ) ) ) ) ) = ( ( D ` M ) .xb .1. ) )
99	19, 24, 98	3eqtr3d 2478	1 |- ( ( M e. B /\ R e. CRing ) -> ( M .x. ( J ` M ) ) = ( ( D ` M ) .xb .1. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   =/= wne 2625   _Vcvv 3087   ifcif 3915   <.cotp 4010   |-> cmpt 4484   X. cxp 4852   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   |-> cmpt2 6307   ^m cmap 7480   Fincfn 7577   Basecbs 15084   .rcmulr 15153   .scvsca 15156   0gc0g 15297   gsum_g cgsu 15298   1rcur 17670   Ringcrg 17715   CRingccrg 17716   maMul cmmul 19339   Mat cmat 19363   maDet cmdat 19540   maAdju cmadu 19588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-addf 9617  ax-mulf 9618

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-xor 1401  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-word 12651  df-lsw 12652  df-concat 12653  df-s1 12654  df-substr 12655  df-splice 12656  df-reverse 12657  df-s2 12929  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-0g 15299  df-gsum 15300  df-prds 15305  df-pws 15307  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-ghm 16832  df-gim 16874  df-cntz 16922  df-oppg 16948  df-symg 16970  df-pmtr 17034  df-psgn 17083  df-evpm 17084  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-cring 17718  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-rnghom 17878  df-drng 17912  df-subrg 17941  df-lmod 18028  df-lss 18091  df-sra 18330  df-rgmod 18331  df-cnfld 18906  df-zring 18974  df-zrh 19006  df-dsmm 19226  df-frlm 19241  df-mamu 19340  df-mat 19364  df-mdet 19541  df-madu 19590

This theorem is referenced by:  madulid  19601  matinv  19633  cpmadurid  19822  cpmidgsum2  19834