Theorem madurid 18941

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 2467	. . 3 |- ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
2		eqid 2467	. . 3 |- ( Base ` R ) = ( Base ` R )
3		eqid 2467	. . 3 |- ( .r ` R ) = ( .r ` R )
4		simpr 461	. . 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 18709	. . . . 5 |- ( M e. B -> ( N e. Fin /\ R e. _V ) )
8	7	simpld 459	. . . 4 |- ( M e. B -> N e. Fin )
9	8	adantr 465	. . 3 |- ( ( M e. B /\ R e. CRing ) -> N e. Fin )
10	5, 2, 6	matbas2i 18719	. . . 4 |- ( M e. B -> M e. ( ( Base ` R ) ^m ( N X. N ) ) )
11	10	adantr 465	. . 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 18938	. . . . . 6 |- ( R e. CRing -> J : B --> B )
14	13	adantl 466	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> J : B --> B )
15		simpl 457	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> M e. B )
16	14, 15	ffvelrnd 6022	. . . 4 |- ( ( M e. B /\ R e. CRing ) -> ( J ` M ) e. B )
17	5, 2, 6	matbas2i 18719	. . . 4 |- ( ( J ` M ) e. B -> ( J ` M ) e. ( ( Base ` R ) ^m ( N X. N ) ) )
18	16, 17	syl 16	. . 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 18683	. 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 18735	. . . . 5 |- ( ( N e. Fin /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = ( .r ` A ) )
21	8, 20	sylan 471	. . . 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 2526	. . 3 |- ( ( M e. B /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = .x. )
24	23	oveqd 6301	. 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 1020	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> M e. B )
27		simp1r 1021	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> R e. CRing )
28		elmapi 7440	. . . . . . . . . 10 |- ( M e. ( ( Base ` R ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` R ) )
29	11, 28	syl 16	. . . . . . . . 9 |- ( ( M e. B /\ R e. CRing ) -> M : ( N X. N ) --> ( Base ` R ) )
30	29	3ad2ant1 1017	. . . . . . . 8 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
31	30	adantr 465	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
32		simpl2 1000	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> a e. N )
33		simpr 461	. . . . . . 7 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> c e. N )
34	31, 32, 33	fovrnd 6431	. . . . . 6 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> ( a M c ) e. ( Base ` R ) )
35		simp3 998	. . . . . 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 18940	. . . . 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 3945	. . . . . . . . 9 |- ( a = b -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( D ` M ) )
38	37	adantl 466	. . . . . . . 8 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( D ` M ) )
39		ffn 5731	. . . . . . . . . . . . 13 |- ( M : ( N X. N ) --> ( Base ` R ) -> M Fn ( N X. N ) )
40	29, 39	syl 16	. . . . . . . . . . . 12 |- ( ( M e. B /\ R e. CRing ) -> M Fn ( N X. N ) )
41		fnov 6394	. . . . . . . . . . . 12 |- ( M Fn ( N X. N ) <-> M = ( d e. N , c e. N |-> ( d M c ) ) )
42	40, 41	sylib 196	. . . . . . . . . . 11 |- ( ( M e. B /\ R e. CRing ) -> M = ( d e. N , c e. N |-> ( d M c ) ) )
43	42	adantr 465	. . . . . . . . . 10 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> M = ( d e. N , c e. N |-> ( d M c ) ) )
44		equtr2 1751	. . . . . . . . . . . . . . 15 |- ( ( a = b /\ d = b ) -> a = d )
45	44	oveq1d 6299	. . . . . . . . . . . . . 14 |- ( ( a = b /\ d = b ) -> ( a M c ) = ( d M c ) )
46	45	ifeq1da 3969	. . . . . . . . . . . . 13 |- ( a = b -> if ( d = b , ( a M c ) , ( d M c ) ) = if ( d = b , ( d M c ) , ( d M c ) ) )
47		ifid 3976	. . . . . . . . . . . . 13 |- if ( d = b , ( d M c ) , ( d M c ) ) = ( d M c )
48	46, 47	syl6eq 2524	. . . . . . . . . . . 12 |- ( a = b -> if ( d = b , ( a M c ) , ( d M c ) ) = ( d M c ) )
49	48	adantl 466	. . . . . . . . . . 11 |- ( ( ( M e. B /\ R e. CRing ) /\ a = b ) -> if ( d = b , ( a M c ) , ( d M c ) ) = ( d M c ) )
50	49	mpt2eq3dv 6347	. . . . . . . . . 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 2511	. . . . . . . . 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 5870	. . . . . . . 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 2509	. . . . . . 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 1158	. . . . . 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 2467	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
56		simpl1r 1048	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> R e. CRing )
57	9	3ad2ant1 1017	. . . . . . . . 9 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> N e. Fin )
58	57	adantr 465	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> N e. Fin )
59	30	ad2antrr 725	. . . . . . . . 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 1036	. . . . . . . . 9 |- ( ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) /\ c e. N ) -> a e. N )
61		simpr 461	. . . . . . . . 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 6431	. . . . . . . 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 465	. . . . . . . . . 10 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> M : ( N X. N ) --> ( Base ` R ) )
64	63	fovrnda 6430	. . . . . . . . 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 1192	. . . . . . . 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 1001	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> b e. N )
67		simpl2 1000	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> a e. N )
68		df-ne 2664	. . . . . . . . . . 11 |- ( a =/= b <-> -. a = b )
69	68	biimpri 206	. . . . . . . . . 10 |- ( -. a = b -> a =/= b )
70	69	necomd 2738	. . . . . . . . 9 |- ( -. a = b -> b =/= a )
71	70	adantl 466	. . . . . . . 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 18906	. . . . . . 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 3976	. . . . . . . . . . 11 |- if ( d = a , ( d M c ) , ( d M c ) ) = ( d M c )
74		oveq1 6291	. . . . . . . . . . . . 13 |- ( d = a -> ( d M c ) = ( a M c ) )
75	74	adantl 466	. . . . . . . . . . . 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 3969	. . . . . . . . . . 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 2522	. . . . . . . . . 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 3958	. . . . . . . . 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 6347	. . . . . . . 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 5870	. . . . . . 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 3948	. . . . . . . 8 |- ( -. a = b -> if ( a = b , ( D ` M ) , ( 0g ` R ) ) = ( 0g ` R ) )
82	81	adantl 466	. . . . . . 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 2518	. . . . . 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 789	. . . . 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 2508	. . . 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 6345	. . 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 6295	. . . 4 |- ( ( D ` M ) .xb .1. ) = ( ( D ` M ) .xb ( 1r ` A ) )
89		crngrng 17010	. . . . . 6 |- ( R e. CRing -> R e. Ring )
90	89	adantl 466	. . . . 5 |- ( ( M e. B /\ R e. CRing ) -> R e. Ring )
91	25, 5, 6, 2	mdetf 18892	. . . . . . 7 |- ( R e. CRing -> D : B --> ( Base ` R ) )
92	91	adantl 466	. . . . . 6 |- ( ( M e. B /\ R e. CRing ) -> D : B --> ( Base ` R ) )
93	92, 15	ffvelrnd 6022	. . . . 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 18747	. . . . 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 1228	. . . 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 2520	. . 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 2511	. 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 2516	1 |- ( ( M e. B /\ R e. CRing ) -> ( M .x. ( J ` M ) ) = ( ( D ` M ) .xb .1. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3113   ifcif 3939   <.cotp 4035   |-> cmpt 4505   X. cxp 4997   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   |-> cmpt2 6286   ^m cmap 7420   Fincfn 7516   Basecbs 14490   .rcmulr 14556   .scvsca 14559   0gc0g 14695   gsum_g cgsu 14696   1rcur 16955   Ringcrg 17000   CRingccrg 17001   maMul cmmul 18680   Mat cmat 18704   maDet cmdat 18881   maAdju cmadu 18929

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 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-addf 9571  ax-mulf 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-fal 1385  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512  df-splice 12513  df-reverse 12514  df-s2 12776  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-0g 14697  df-gsum 14698  df-prds 14703  df-pws 14705  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-ghm 16070  df-gim 16112  df-cntz 16160  df-oppg 16186  df-symg 16208  df-pmtr 16273  df-psgn 16322  df-evpm 16323  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-subrg 17227  df-lmod 17314  df-lss 17379  df-sra 17618  df-rgmod 17619  df-cnfld 18220  df-zring 18285  df-zrh 18336  df-dsmm 18558  df-frlm 18573  df-mamu 18681  df-mat 18705  df-mdet 18882  df-madu 18931

This theorem is referenced by:  madulid  18942  matinv  18974  cpmadurid  19163  cpmidgsum2  19175