Theorem pmatcollpwlem 20585

Description: Lemma for pmatcollpw 20586. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpw.p	|- P = (Poly1 ` R )
pmatcollpw.c	|- C = ( N Mat P )
pmatcollpw.b	|- B = ( Base ` C )
pmatcollpw.m	|- .* = ( .s ` C )
pmatcollpw.e	|- .^ = (.g ` (mulGrp ` P ) )
pmatcollpw.x	|- X = (var1 ` R )
pmatcollpw.t	|- T = ( N matToPolyMat R )

Assertion

Ref	Expression
pmatcollpwlem	|- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( ( a ( M decompPMat n ) b ) ( .s ` P ) ( n .^ X ) ) = ( a ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) b ) )

Proof of Theorem pmatcollpwlem

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmatcollpw.p	. . . . . . . 8 |- P = (Poly1 ` R )
2	1	ply1assa 19569	. . . . . . 7 |- ( R e. CRing -> P e. AssAlg )
3	2	3ad2ant2 1083	. . . . . 6 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> P e. AssAlg )
4	3	adantr 481	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> P e. AssAlg )
5	4	3ad2ant1 1082	. . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> P e. AssAlg )
6		eqid 2622	. . . . . 6 |- ( N Mat R ) = ( N Mat R )
7		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
8		eqid 2622	. . . . . 6 |- ( Base ` ( N Mat R ) ) = ( Base ` ( N Mat R ) )
9		simp2 1062	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> a e. N )
10		simp3 1063	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> b e. N )
11		simp2 1062	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> R e. CRing )
12	11	adantr 481	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> R e. CRing )
13		simp3 1063	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> M e. B )
14	13	adantr 481	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> M e. B )
15		simpr 477	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> n e. NN0 )
16		pmatcollpw.c	. . . . . . . . 9 |- C = ( N Mat P )
17		pmatcollpw.b	. . . . . . . . 9 |- B = ( Base ` C )
18	1, 16, 17, 6, 8	decpmatcl 20572	. . . . . . . 8 |- ( ( R e. CRing /\ M e. B /\ n e. NN0 ) -> ( M decompPMat n ) e. ( Base ` ( N Mat R ) ) )
19	12, 14, 15, 18	syl3anc 1326	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( M decompPMat n ) e. ( Base ` ( N Mat R ) ) )
20	19	3ad2ant1 1082	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( M decompPMat n ) e. ( Base ` ( N Mat R ) ) )
21	6, 7, 8, 9, 10, 20	matecld 20232	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( a ( M decompPMat n ) b ) e. ( Base ` R ) )
22		crngring 18558	. . . . . . . . . . . 12 |- ( R e. CRing -> R e. Ring )
23	22	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> R e. Ring )
24	1	ply1sca 19623	. . . . . . . . . . 11 |- ( R e. Ring -> R = (Scalar ` P ) )
25	23, 24	syl 17	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> R = (Scalar ` P ) )
26	25	eqcomd 2628	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> (Scalar ` P ) = R )
27	26	fveq2d 6195	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
28	27	eleq2d 2687	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( a ( M decompPMat n ) b ) e. ( Base ` (Scalar ` P ) ) <-> ( a ( M decompPMat n ) b ) e. ( Base ` R ) ) )
29	28	adantr 481	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( ( a ( M decompPMat n ) b ) e. ( Base ` (Scalar ` P ) ) <-> ( a ( M decompPMat n ) b ) e. ( Base ` R ) ) )
30	29	3ad2ant1 1082	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( ( a ( M decompPMat n ) b ) e. ( Base ` (Scalar ` P ) ) <-> ( a ( M decompPMat n ) b ) e. ( Base ` R ) ) )
31	21, 30	mpbird 247	. . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( a ( M decompPMat n ) b ) e. ( Base ` (Scalar ` P ) ) )
32		pmatcollpw.x	. . . . . . 7 |- X = (var1 ` R )
33		eqid 2622	. . . . . . 7 |- (mulGrp ` P ) = (mulGrp ` P )
34		pmatcollpw.e	. . . . . . 7 |- .^ = (.g ` (mulGrp ` P ) )
35		eqid 2622	. . . . . . 7 |- ( Base ` P ) = ( Base ` P )
36	1, 32, 33, 34, 35	ply1moncl 19641	. . . . . 6 |- ( ( R e. Ring /\ n e. NN0 ) -> ( n .^ X ) e. ( Base ` P ) )
37	23, 36	sylan 488	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( n .^ X ) e. ( Base ` P ) )
38	37	3ad2ant1 1082	. . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( n .^ X ) e. ( Base ` P ) )
39		eqid 2622	. . . . 5 |- (algSc ` P ) = (algSc ` P )
40		eqid 2622	. . . . 5 |- (Scalar ` P ) = (Scalar ` P )
41		eqid 2622	. . . . 5 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
42		eqid 2622	. . . . 5 |- ( .r ` P ) = ( .r ` P )
43		eqid 2622	. . . . 5 |- ( .s ` P ) = ( .s ` P )
44	39, 40, 41, 35, 42, 43	asclmul2 19340	. . . 4 |- ( ( P e. AssAlg /\ ( a ( M decompPMat n ) b ) e. ( Base ` (Scalar ` P ) ) /\ ( n .^ X ) e. ( Base ` P ) ) -> ( ( n .^ X ) ( .r ` P ) ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) ) = ( ( a ( M decompPMat n ) b ) ( .s ` P ) ( n .^ X ) ) )
45	5, 31, 38, 44	syl3anc 1326	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( ( n .^ X ) ( .r ` P ) ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) ) = ( ( a ( M decompPMat n ) b ) ( .s ` P ) ( n .^ X ) ) )
46		eqidd 2623	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) = ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) )
47		oveq12 6659	. . . . . . . 8 |- ( ( i = a /\ j = b ) -> ( i ( M decompPMat n ) j ) = ( a ( M decompPMat n ) b ) )
48	47	fveq2d 6195	. . . . . . 7 |- ( ( i = a /\ j = b ) -> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) = ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) )
49	48	adantl 482	. . . . . 6 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) /\ ( i = a /\ j = b ) ) -> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) = ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) )
50		fvexd 6203	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) e. _V )
51	46, 49, 9, 10, 50	ovmpt2d 6788	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( a ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) b ) = ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) )
52	51	eqcomd 2628	. . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) = ( a ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) b ) )
53	52	oveq2d 6666	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( ( n .^ X ) ( .r ` P ) ( (algSc ` P ) ` ( a ( M decompPMat n ) b ) ) ) = ( ( n .^ X ) ( .r ` P ) ( a ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) b ) ) )
54	45, 53	eqtr3d 2658	. 2 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( ( a ( M decompPMat n ) b ) ( .s ` P ) ( n .^ X ) ) = ( ( n .^ X ) ( .r ` P ) ( a ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) b ) ) )
55	1	ply1ring 19618	. . . . . . 7 |- ( R e. Ring -> P e. Ring )
56	22, 55	syl 17	. . . . . 6 |- ( R e. CRing -> P e. Ring )
57	56	3ad2ant2 1083	. . . . 5 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> P e. Ring )
58	57	adantr 481	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> P e. Ring )
59	58	3ad2ant1 1082	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> P e. Ring )
60		simpl1 1064	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> N e. Fin )
61	12, 22	syl 17	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> R e. Ring )
62	61	3ad2ant1 1082	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ i e. N /\ j e. N ) -> R e. Ring )
63		simp2 1062	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ i e. N /\ j e. N ) -> i e. N )
64		simp3 1063	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ i e. N /\ j e. N ) -> j e. N )
65	19	3ad2ant1 1082	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ i e. N /\ j e. N ) -> ( M decompPMat n ) e. ( Base ` ( N Mat R ) ) )
66	6, 7, 8, 63, 64, 65	matecld 20232	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ i e. N /\ j e. N ) -> ( i ( M decompPMat n ) j ) e. ( Base ` R ) )
67	1, 39, 7, 35	ply1sclcl 19656	. . . . . . 7 |- ( ( R e. Ring /\ ( i ( M decompPMat n ) j ) e. ( Base ` R ) ) -> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) e. ( Base ` P ) )
68	62, 66, 67	syl2anc 693	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ i e. N /\ j e. N ) -> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) e. ( Base ` P ) )
69	16, 35, 17, 60, 58, 68	matbas2d 20229	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) e. B )
70	37, 69	jca 554	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( ( n .^ X ) e. ( Base ` P ) /\ ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) e. B ) )
71	70	3ad2ant1 1082	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( ( n .^ X ) e. ( Base ` P ) /\ ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) e. B ) )
72	9, 10	jca 554	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( a e. N /\ b e. N ) )
73		pmatcollpw.m	. . . 4 |- .* = ( .s ` C )
74	16, 17, 35, 73, 42	matvscacell 20242	. . 3 |- ( ( P e. Ring /\ ( ( n .^ X ) e. ( Base ` P ) /\ ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) e. B ) /\ ( a e. N /\ b e. N ) ) -> ( a ( ( n .^ X ) .* ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) ) b ) = ( ( n .^ X ) ( .r ` P ) ( a ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) b ) ) )
75	59, 71, 72, 74	syl3anc 1326	. 2 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( a ( ( n .^ X ) .* ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) ) b ) = ( ( n .^ X ) ( .r ` P ) ( a ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) b ) ) )
76	23	adantr 481	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> R e. Ring )
77		pmatcollpw.t	. . . . . . . 8 |- T = ( N matToPolyMat R )
78	77, 6, 8, 1, 39	mat2pmatval 20529	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ ( M decompPMat n ) e. ( Base ` ( N Mat R ) ) ) -> ( T ` ( M decompPMat n ) ) = ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) )
79	60, 76, 19, 78	syl3anc 1326	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( T ` ( M decompPMat n ) ) = ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) )
80	79	eqcomd 2628	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) = ( T ` ( M decompPMat n ) ) )
81	80	oveq2d 6666	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( ( n .^ X ) .* ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) ) = ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) )
82	81	oveqd 6667	. . 3 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) -> ( a ( ( n .^ X ) .* ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) ) b ) = ( a ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) b ) )
83	82	3ad2ant1 1082	. 2 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( a ( ( n .^ X ) .* ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat n ) j ) ) ) ) b ) = ( a ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) b ) )
84	54, 75, 83	3eqtr2d 2662	1 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ n e. NN0 ) /\ a e. N /\ b e. N ) -> ( ( a ( M decompPMat n ) b ) ( .s ` P ) ( n .^ X ) ) = ( a ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) b ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   NN0cn0 11292   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548  AssAlgcasa 19309  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547   Mat cmat 20213   matToPolyMat cmat2pmat 20509   decompPMat cdecpmat 20567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-assa 19312  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-dsmm 20076  df-frlm 20091  df-mat 20214  df-mat2pmat 20512  df-decpmat 20568

This theorem is referenced by:  pmatcollpw  20586