Theorem madurid 19669

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 2451	. . 3 |- ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
2		eqid 2451	. . 3 |- ( Base ` R ) = ( Base ` R )
3		eqid 2451	. . 3 |- ( .r ` R ) = ( .r ` R )
4		simpr 463	. . 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 19437	. . . . 5 |- ( M e. B -> ( N e. Fin /\ R e. _V ) )
8	7	simpld 461	. . . 4 |- ( M e. B -> N e. Fin )
9	8	adantr 467	. . 3 |- ( ( M e. B /\ R e. CRing ) -> N e. Fin )
10	5, 2, 6	matbas2i 19447	. . . 4 |- ( M e. B -> M e. ( ( Base ` R ) ^m ( N X. N ) ) )
11	10	adantr 467	. . 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 19666	. . . . . 6 |- ( R e. CRing -> J : B --> B )
14	13	adantl 468	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> J : B --> B )
15		simpl 459	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> M e. B )
16	14, 15	ffvelrnd 6023	. . . 4 |- ( ( M e. B /\ R e. CRing ) -> ( J ` M ) e. B )
17	5, 2, 6	matbas2i 19447	. . . 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 19411	. 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 19463	. . . . 5 |- ( ( N e. Fin /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = ( .r ` A ) )
21	8, 20	sylan 474	. . . 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 2503	. . 3 |- ( ( M e. B /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = .x. )
24	23	oveqd 6307	. 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 1032	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> M e. B )
27		simp1r 1033	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> R e. CRing )
28		elmapi 7493	. . . . . . . . . 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 1029	. . . . . . . 8 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
31	30	adantr 467	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
32		simpl2 1012	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> a e. N )
33		simpr 463	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> c e. N )
34	31, 32, 33	fovrnd 6441	. . . . . 6 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> ( a M c ) e. ( Base ` R ) )
35		simp3 1010	. . . . . 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 19668	. . . . 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 3887	. . . . . . . . 9 |- ( a = b -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( D ` M ) )
38	37	adantl 468	. . . . . . . 8 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( D ` M ) )
39		ffn 5728	. . . . . . . . . . . . 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 6404	. . . . . . . . . . . 12 |- ( M Fn ( N X. N ) <-> M = ( d e. N , c e. N |-> ( d M c ) ) )
42	40, 41	sylib 200	. . . . . . . . . . 11 |- ( ( M e. B /\ R e. CRing ) -> M = ( d e. N , c e. N |-> ( d M c ) ) )
43	42	adantr 467	. . . . . . . . . 10 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> M = ( d e. N , c e. N |-> ( d M c ) ) )
44		equtr2 1869	. . . . . . . . . . . . . . 15 |- ( ( a = b /\ d = b ) -> a = d )
45	44	oveq1d 6305	. . . . . . . . . . . . . 14 |- ( ( a = b /\ d = b ) -> ( a M c ) = ( d M c ) )
46	45	ifeq1da 3911	. . . . . . . . . . . . 13 |- ( a = b -> if ( d = b , ( a M c ) , ( d M c ) ) = if ( d = b , ( d M c ) , ( d M c ) ) )
47		ifid 3918	. . . . . . . . . . . . 13 |- if ( d = b , ( d M c ) , ( d M c ) ) = ( d M c )
48	46, 47	syl6eq 2501	. . . . . . . . . . . 12 |- ( a = b -> if ( d = b , ( a M c ) , ( d M c ) ) = ( d M c ) )
49	48	adantl 468	. . . . . . . . . . 11 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> if ( d = b , ( a M c ) , ( d M c ) ) = ( d M c ) )
50	49	mpt2eq3dv 6357	. . . . . . . . . 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 2488	. . . . . . . . 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 5869	. . . . . . . 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 2486	. . . . . . 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 1170	. . . . . 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 2451	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
56		simpl1r 1060	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> R e. CRing )
57	9	3ad2ant1 1029	. . . . . . . . 9 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> N e. Fin )
58	57	adantr 467	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> N e. Fin )
59	30	ad2antrr 732	. . . . . . . . 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 1048	. . . . . . . . 9 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ c e. N ) -> a e. N )
61		simpr 463	. . . . . . . . 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 6441	. . . . . . . 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 467	. . . . . . . . . 10 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> M : ( N X. N ) --> ( Base ` R ) )
64	63	fovrnda 6440	. . . . . . . . 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 1204	. . . . . . . 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 1013	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> b e. N )
67		simpl2 1012	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> a e. N )
68		df-ne 2624	. . . . . . . . . . 11 |- ( a =/= b <-> -. a = b )
69	68	biimpri 210	. . . . . . . . . 10 |- ( -. a = b -> a =/= b )
70	69	necomd 2679	. . . . . . . . 9 |- ( -. a = b -> b =/= a )
71	70	adantl 468	. . . . . . . 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 19634	. . . . . . 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 3918	. . . . . . . . . . 11 |- if ( d = a , ( d M c ) , ( d M c ) ) = ( d M c )
74		oveq1 6297	. . . . . . . . . . . . 13 |- ( d = a -> ( d M c ) = ( a M c ) )
75	74	adantl 468	. . . . . . . . . . . 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 3911	. . . . . . . . . . 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 2499	. . . . . . . . . 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 3900	. . . . . . . . 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 6357	. . . . . . . 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 5869	. . . . . . 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 3890	. . . . . . . 8 |- ( -. a = b -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( 0g ` R ) )
82	81	adantl 468	. . . . . . 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 2495	. . . . . 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 800	. . . . 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 2485	. . . 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 6355	. . 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 6301	. . . 4 |- ( ( D ` M ) .xb .1. ) = ( ( D ` M ) .xb ( 1r ` A ) )
89		crngring 17791	. . . . . 6 |- ( R e. CRing -> R e. Ring )
90	89	adantl 468	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> R e. Ring )
91	25, 5, 6, 2	mdetf 19620	. . . . . . 7 |- ( R e. CRing -> D : B --> ( Base ` R ) )
92	91	adantl 468	. . . . . 6 |- ( ( M e. B /\ R e. CRing ) -> D : B --> ( Base ` R ) )
93	92, 15	ffvelrnd 6023	. . . . 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 19475	. . . . 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 1268	. . . 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 2497	. . 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 2488	. 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 2493	1 |- ( ( M e. B /\ R e. CRing ) -> ( M .x. ( J ` M ) ) = ( ( D ` M ) .xb .1. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   ifcif 3881   <.cotp 3976   |-> cmpt 4461   X. cxp 4832   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   ^m cmap 7472   Fincfn 7569   Basecbs 15121   .rcmulr 15191   .scvsca 15194   0gc0g 15338   gsum_g cgsu 15339   1rcur 17735   Ringcrg 17780   CRingccrg 17781   maMul cmmul 19408   Mat cmat 19432   maDet cmdat 19609   maAdju cmadu 19657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-xor 1406  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  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 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-splice 12669  df-reverse 12670  df-s2 12944  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-0g 15340  df-gsum 15341  df-prds 15346  df-pws 15348  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-gim 16923  df-cntz 16971  df-oppg 16997  df-symg 17019  df-pmtr 17083  df-psgn 17132  df-evpm 17133  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-rnghom 17943  df-drng 17977  df-subrg 18006  df-lmod 18093  df-lss 18156  df-sra 18395  df-rgmod 18396  df-cnfld 18971  df-zring 19040  df-zrh 19075  df-dsmm 19295  df-frlm 19310  df-mamu 19409  df-mat 19433  df-mdet 19610  df-madu 19659

This theorem is referenced by:  madulid  19670  matinv  19702  cpmadurid  19891  cpmidgsum2  19903