Theorem mp2pm2mp 20616

Description: A polynomial over matrices transformed into a polynomial matrix transformed back into the polynomial over matrices. (Contributed by AV, 12-Oct-2019.)

Hypotheses

Ref	Expression
mp2pm2mp.a	|- A = ( N Mat R )
mp2pm2mp.q	|- Q = (Poly1 ` A )
mp2pm2mp.l	|- L = ( Base ` Q )
mp2pm2mp.m	|- .x. = ( .s ` P )
mp2pm2mp.e	|- E = (.g ` (mulGrp ` P ) )
mp2pm2mp.y	|- Y = (var1 ` R )
mp2pm2mp.i	|- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )
mp2pm2mplem2.p	|- P = (Poly1 ` R )
mp2pm2mp.t	|- T = ( N pMatToMatPoly R )

Assertion

Ref	Expression
mp2pm2mp	|- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( T ` ( I ` O ) ) = O )

Distinct variable groups:   E, p   L, p   i, N, j, p   i, O, j, p, k   P, p   R, p   Y, p   .x. , p   k, L   P, i, j, k   R, k   .x. , k   i, E, j   i, L, j   k, N   R, i, j   i, Y, j   .x. , i, j   A, i, j, k   k, E   k, Y

Allowed substitution hints:   A( p)   Q( i, j, k, p)   T( i, j, k, p)   I( i, j, k, p)

Proof of Theorem mp2pm2mp

Dummy variables n l are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mp2pm2mp.a	. . . 4 |- A = ( N Mat R )
2		mp2pm2mp.q	. . . 4 |- Q = (Poly1 ` A )
3		mp2pm2mp.l	. . . 4 |- L = ( Base ` Q )
4		mp2pm2mplem2.p	. . . 4 |- P = (Poly1 ` R )
5		mp2pm2mp.m	. . . 4 |- .x. = ( .s ` P )
6		mp2pm2mp.e	. . . 4 |- E = (.g ` (mulGrp ` P ) )
7		mp2pm2mp.y	. . . 4 |- Y = (var1 ` R )
8		mp2pm2mp.i	. . . 4 |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) )
9		eqid 2622	. . . 4 |- ( N Mat P ) = ( N Mat P )
10		eqid 2622	. . . 4 |- ( Base ` ( N Mat P ) ) = ( Base ` ( N Mat P ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	mply1topmatcl 20610	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( I ` O ) e. ( Base ` ( N Mat P ) ) )
12		eqid 2622	. . . 4 |- ( .s ` Q ) = ( .s ` Q )
13		eqid 2622	. . . 4 |- (.g ` (mulGrp ` Q ) ) = (.g ` (mulGrp ` Q ) )
14		eqid 2622	. . . 4 |- (var1 ` A ) = (var1 ` A )
15		mp2pm2mp.t	. . . 4 |- T = ( N pMatToMatPoly R )
16	4, 9, 10, 12, 13, 14, 1, 2, 15	pm2mpfval 20601	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ ( I ` O ) e. ( Base ` ( N Mat P ) ) ) -> ( T ` ( I ` O ) ) = ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
17	11, 16	syld3an3 1371	. 2 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( T ` ( I ` O ) ) = ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
18	1	matring 20249	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring )
19	18	3adant3 1081	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> A e. Ring )
20		eqid 2622	. . . . 5 |- ( 0g ` Q ) = ( 0g ` Q )
21	2	ply1ring 19618	. . . . . . 7 |- ( A e. Ring -> Q e. Ring )
22		ringcmn 18581	. . . . . . 7 |- ( Q e. Ring -> Q e. CMnd )
23	18, 21, 22	3syl 18	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> Q e. CMnd )
24	23	3adant3 1081	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> Q e. CMnd )
25		nn0ex 11298	. . . . . 6 |- NN0 e. _V
26	25	a1i 11	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> NN0 e. _V )
27	19	adantr 481	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> A e. Ring )
28		simpl2 1065	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> R e. Ring )
29	11	adantr 481	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> ( I ` O ) e. ( Base ` ( N Mat P ) ) )
30		simpr 477	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> n e. NN0 )
31		eqid 2622	. . . . . . . . 9 |- ( Base ` A ) = ( Base ` A )
32	4, 9, 10, 1, 31	decpmatcl 20572	. . . . . . . 8 |- ( ( R e. Ring /\ ( I ` O ) e. ( Base ` ( N Mat P ) ) /\ n e. NN0 ) -> ( ( I ` O ) decompPMat n ) e. ( Base ` A ) )
33	28, 29, 30, 32	syl3anc 1326	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> ( ( I ` O ) decompPMat n ) e. ( Base ` A ) )
34		eqid 2622	. . . . . . . 8 |- (mulGrp ` Q ) = (mulGrp ` Q )
35	31, 2, 14, 12, 34, 13, 3	ply1tmcl 19642	. . . . . . 7 |- ( ( A e. Ring /\ ( ( I ` O ) decompPMat n ) e. ( Base ` A ) /\ n e. NN0 ) -> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. L )
36	27, 33, 30, 35	syl3anc 1326	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. L )
37		eqid 2622	. . . . . 6 |- ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) = ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) )
38	36, 37	fmptd 6385	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) : NN0 --> L )
39		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = n -> ( (coe1 ` p ) ` k ) = ( (coe1 ` p ) ` n ) )
40	39	oveqd 6667	. . . . . . . . . . . . 13 |- ( k = n -> ( i ( (coe1 ` p ) ` k ) j ) = ( i ( (coe1 ` p ) ` n ) j ) )
41		oveq1 6657	. . . . . . . . . . . . 13 |- ( k = n -> ( k E Y ) = ( n E Y ) )
42	40, 41	oveq12d 6668	. . . . . . . . . . . 12 |- ( k = n -> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) = ( ( i ( (coe1 ` p ) ` n ) j ) .x. ( n E Y ) ) )
43	42	cbvmptv 4750	. . . . . . . . . . 11 |- ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) = ( n e. NN0 |-> ( ( i ( (coe1 ` p ) ` n ) j ) .x. ( n E Y ) ) )
44	43	a1i 11	. . . . . . . . . 10 |- ( ( i e. N /\ j e. N ) -> ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) = ( n e. NN0 |-> ( ( i ( (coe1 ` p ) ` n ) j ) .x. ( n E Y ) ) ) )
45	44	oveq2d 6666	. . . . . . . . 9 |- ( ( i e. N /\ j e. N ) -> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) = ( P gsumg ( n e. NN0 |-> ( ( i ( (coe1 ` p ) ` n ) j ) .x. ( n E Y ) ) ) ) )
46	45	mpt2eq3ia 6720	. . . . . . . 8 |- ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) = ( i e. N , j e. N |-> ( P gsumg ( n e. NN0 |-> ( ( i ( (coe1 ` p ) ` n ) j ) .x. ( n E Y ) ) ) ) )
47	46	mpteq2i 4741	. . . . . . 7 |- ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) ) ) ) ) = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( n e. NN0 |-> ( ( i ( (coe1 ` p ) ` n ) j ) .x. ( n E Y ) ) ) ) ) )
48	8, 47	eqtri 2644	. . . . . 6 |- I = ( p e. L |-> ( i e. N , j e. N |-> ( P gsumg ( n e. NN0 |-> ( ( i ( (coe1 ` p ) ` n ) j ) .x. ( n E Y ) ) ) ) ) )
49	1, 2, 3, 5, 6, 7, 48, 4, 12, 13, 14	mp2pm2mplem5 20615	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
50	3, 20, 24, 26, 38, 49	gsumcl 18316	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. L )
51		simp3 1063	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> O e. L )
52	19, 50, 51	3jca 1242	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( A e. Ring /\ ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. L /\ O e. L ) )
53	1, 2, 3, 5, 6, 7, 8, 4	mp2pm2mplem4 20614	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> ( ( I ` O ) decompPMat n ) = ( (coe1 ` O ) ` n ) )
54	53	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ n e. NN0 ) -> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) = ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) )
55	54	adantlr 751	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) /\ n e. NN0 ) -> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) = ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) )
56	55	mpteq2dva 4744	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) = ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) )
57	56	oveq2d 6666	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
58	57	fveq2d 6195	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) = (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) )
59	58	fveq1d 6193	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> ( (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) = ( (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) )
60	19, 51	jca 554	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( A e. Ring /\ O e. L ) )
61	60	adantr 481	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> ( A e. Ring /\ O e. L ) )
62		eqid 2622	. . . . . . . . . 10 |- (coe1 ` O ) = (coe1 ` O )
63	2, 14, 3, 12, 34, 13, 62	ply1coe 19666	. . . . . . . . 9 |- ( ( A e. Ring /\ O e. L ) -> O = ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
64	61, 63	syl 17	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> O = ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
65	64	eqcomd 2628	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = O )
66	65	fveq2d 6195	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) = (coe1 ` O ) )
67	66	fveq1d 6193	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> ( (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( (coe1 ` O ) ` n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) = ( (coe1 ` O ) ` l ) )
68	59, 67	eqtrd 2656	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ O e. L ) /\ l e. NN0 ) -> ( (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) = ( (coe1 ` O ) ` l ) )
69	68	ralrimiva 2966	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> A. l e. NN0 ( (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) = ( (coe1 ` O ) ` l ) )
70		eqid 2622	. . . 4 |- (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) = (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
71	2, 3, 70, 62	eqcoe1ply1eq 19667	. . 3 |- ( ( A e. Ring /\ ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. L /\ O e. L ) -> ( A. l e. NN0 ( (coe1 ` ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) = ( (coe1 ` O ) ` l ) -> ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = O ) )
72	52, 69, 71	sylc 65	. 2 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( Q gsumg ( n e. NN0 |-> ( ( ( I ` O ) decompPMat n ) ( .s ` Q ) ( n (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = O )
73	17, 72	eqtrd 2656	1 |- ( ( N e. Fin /\ R e. Ring /\ O e. L ) -> ( T ` ( I ` O ) ) = O )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   NN0cn0 11292   Basecbs 15857   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101  ._gcmg 17540  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   decompPMat cdecpmat 20567   pMatToMatPoly cpm2mp 20597

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-fal 1489  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-srg 18506  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  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-mamu 20190  df-mat 20214  df-decpmat 20568  df-pm2mp 20598

This theorem is referenced by:  pm2mpfo  20619