Theorem madurid 18462

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 2443	. . 3 |- ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
2		eqid 2443	. . 3 |- ( Base ` R ) = ( Base ` R )
3		eqid 2443	. . 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 18324	. . . . 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 18335	. . . 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 18459	. . . . . 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 5856	. . . 4 |- ( ( M e. B /\ R e. CRing ) -> ( J ` M ) e. B )
17	5, 2, 6	matbas2i 18335	. . . 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 18296	. 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 18325	. . . . 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 2493	. . 3 |- ( ( M e. B /\ R e. CRing ) -> ( R maMul <. N , N , N >. ) = .x. )
24	23	oveqd 6120	. 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 1012	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> M e. B )
27		simp1r 1013	. . . . . 6 |- ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) -> R e. CRing )
28		elmapi 7246	. . . . . . . . . 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 1009	. . . . . . . 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 992	. . . . . . 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 6247	. . . . . 6 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ c e. N ) -> ( a M c ) e. ( Base ` R ) )
35		simp3 990	. . . . . 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 18461	. . . . 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 3809	. . . . . . . . 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 5571	. . . . . . . . . . . . 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 6210	. . . . . . . . . . . 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 1740	. . . . . . . . . . . . . . 15 |- ( ( a = b /\ d = b ) -> a = d )
45	44	oveq1d 6118	. . . . . . . . . . . . . 14 |- ( ( a = b /\ d = b ) -> ( a M c ) = ( d M c ) )
46	45	ifeq1da 3831	. . . . . . . . . . . . 13 |- ( a = b -> if ( d = b , ( a M c ) , ( d M c ) ) = if ( d = b , ( d M c ) , ( d M c ) ) )
47		ifid 3838	. . . . . . . . . . . . 13 |- if ( d = b , ( d M c ) , ( d M c ) ) = ( d M c )
48	46, 47	syl6eq 2491	. . . . . . . . . . . 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 6164	. . . . . . . . . 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 2478	. . . . . . . . 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 5707	. . . . . . . 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 2476	. . . . . . 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 1150	. . . . . 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 2443	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
56		simpl1r 1040	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> R e. CRing )
57	9	3ad2ant1 1009	. . . . . . . . 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 1028	. . . . . . . . 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 6247	. . . . . . . 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 6246	. . . . . . . . 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 1183	. . . . . . . 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 993	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> b e. N )
67		simpl2 992	. . . . . . . 8 |- ( ( ( ( M e. B /\ R e. CRing ) /\ a e. N /\ b e. N ) /\ -. a = b ) -> a e. N )
68		df-ne 2620	. . . . . . . . . . 11 |- ( a =/= b <-> -. a = b )
69	68	biimpri 206	. . . . . . . . . 10 |- ( -. a = b -> a =/= b )
70	69	necomd 2707	. . . . . . . . 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 18427	. . . . . . 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 3838	. . . . . . . . . . 11 |- if ( d = a , ( d M c ) , ( d M c ) ) = ( d M c )
74		oveq1 6110	. . . . . . . . . . . . 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 3831	. . . . . . . . . . 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 2489	. . . . . . . . . 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 3820	. . . . . . . . 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 6164	. . . . . . . 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 5707	. . . . . . 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 3811	. . . . . . . 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 2485	. . . . . 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 2475	. . . 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 6162	. . 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 6114	. . . 4 |- ( ( D ` M ) .xb .1. ) = ( ( D ` M ) .xb ( 1r ` A ) )
89		crngrng 16667	. . . . . 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 18418	. . . . . . 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 5856	. . . . 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 18353	. . . . 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 1218	. . . 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 2487	. . 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 2478	. 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 2483	1 |- ( ( M e. B /\ R e. CRing ) -> ( M .x. ( J ` M ) ) = ( ( D ` M ) .xb .1. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2618   _Vcvv 2984   ifcif 3803   <.cotp 3897   e. cmpt 4362   X. cxp 4850   Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   e. cmpt2 6105   ^m cmap 7226   Fincfn 7322   Basecbs 14186   .rcmulr 14251   .scvsca 14254   0gc0g 14390   gsum_g cgsu 14391   1rcur 16615   Ringcrg 16657   CRingccrg 16658   maMul cmmul 18291   Mat cmat 18292   maDet cmdat 18407   maAdju cmadu 18450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-addf 9373  ax-mulf 9374

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-ot 3898  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-tpos 6757  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-seq 11819  df-exp 11878  df-hash 12116  df-word 12241  df-concat 12243  df-s1 12244  df-substr 12245  df-splice 12246  df-reverse 12247  df-s2 12487  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-0g 14392  df-gsum 14393  df-prds 14398  df-pws 14400  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-subg 15690  df-ghm 15757  df-gim 15799  df-cntz 15847  df-oppg 15873  df-symg 15895  df-pmtr 15960  df-psgn 16009  df-evpm 16010  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-cring 16660  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-dvr 16787  df-rnghom 16818  df-drng 16846  df-subrg 16875  df-lmod 16962  df-lss 17026  df-sra 17265  df-rgmod 17266  df-cnfld 17831  df-zring 17896  df-zrh 17947  df-dsmm 18169  df-frlm 18184  df-mamu 18293  df-mat 18294  df-mdet 18408  df-madu 18452

This theorem is referenced by:  madulid  18463  matinv  18495