Theorem pmatcollpwscmatlem2 20595

Description: Lemma 2 for pmatcollpwscmat 20596. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpwscmat.p	|- P = (Poly1 ` R )
pmatcollpwscmat.c	|- C = ( N Mat P )
pmatcollpwscmat.b	|- B = ( Base ` C )
pmatcollpwscmat.m1	|- .* = ( .s ` C )
pmatcollpwscmat.e1	|- .^ = (.g ` (mulGrp ` P ) )
pmatcollpwscmat.x	|- X = (var1 ` R )
pmatcollpwscmat.t	|- T = ( N matToPolyMat R )
pmatcollpwscmat.a	|- A = ( N Mat R )
pmatcollpwscmat.d	|- D = ( Base ` A )
pmatcollpwscmat.u	|- U = (algSc ` P )
pmatcollpwscmat.k	|- K = ( Base ` R )
pmatcollpwscmat.e2	|- E = ( Base ` P )
pmatcollpwscmat.s	|- S = (algSc ` P )
pmatcollpwscmat.1	|- .1. = ( 1r ` C )
pmatcollpwscmat.m2	|- M = ( Q .* .1. )

Assertion

Ref	Expression
pmatcollpwscmatlem2	|- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( T ` ( M decompPMat L ) ) = ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) )

Proof of Theorem pmatcollpwscmatlem2

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( N e. Fin /\ R e. Ring ) )
2		simpr 477	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> R e. Ring )
3	2	adantr 481	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> R e. Ring )
4		simpr 477	. . . . . . . 8 |- ( ( L e. NN0 /\ Q e. E ) -> Q e. E )
5	4	anim2i 593	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( N e. Fin /\ R e. Ring ) /\ Q e. E ) )
6		df-3an 1039	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ Q e. E ) <-> ( ( N e. Fin /\ R e. Ring ) /\ Q e. E ) )
7	5, 6	sylibr 224	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( N e. Fin /\ R e. Ring /\ Q e. E ) )
8		pmatcollpwscmat.m2	. . . . . . 7 |- M = ( Q .* .1. )
9		pmatcollpwscmat.p	. . . . . . . 8 |- P = (Poly1 ` R )
10		pmatcollpwscmat.c	. . . . . . . 8 |- C = ( N Mat P )
11		pmatcollpwscmat.b	. . . . . . . 8 |- B = ( Base ` C )
12		pmatcollpwscmat.e2	. . . . . . . 8 |- E = ( Base ` P )
13		pmatcollpwscmat.m1	. . . . . . . 8 |- .* = ( .s ` C )
14		pmatcollpwscmat.1	. . . . . . . 8 |- .1. = ( 1r ` C )
15	9, 10, 11, 12, 13, 14	1pmatscmul 20507	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ Q e. E ) -> ( Q .* .1. ) e. B )
16	8, 15	syl5eqel 2705	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ Q e. E ) -> M e. B )
17	7, 16	syl 17	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> M e. B )
18		simprl 794	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> L e. NN0 )
19		pmatcollpwscmat.a	. . . . . 6 |- A = ( N Mat R )
20		pmatcollpwscmat.d	. . . . . 6 |- D = ( Base ` A )
21	9, 10, 11, 19, 20	decpmatcl 20572	. . . . 5 |- ( ( R e. Ring /\ M e. B /\ L e. NN0 ) -> ( M decompPMat L ) e. D )
22	3, 17, 18, 21	syl3anc 1326	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( M decompPMat L ) e. D )
23		df-3an 1039	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ ( M decompPMat L ) e. D ) <-> ( ( N e. Fin /\ R e. Ring ) /\ ( M decompPMat L ) e. D ) )
24	1, 22, 23	sylanbrc 698	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( N e. Fin /\ R e. Ring /\ ( M decompPMat L ) e. D ) )
25		pmatcollpwscmat.t	. . . 4 |- T = ( N matToPolyMat R )
26		eqid 2622	. . . 4 |- (algSc ` P ) = (algSc ` P )
27	25, 19, 20, 9, 26	mat2pmatval 20529	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ ( M decompPMat L ) e. D ) -> ( T ` ( M decompPMat L ) ) = ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat L ) j ) ) ) )
28	24, 27	syl 17	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( T ` ( M decompPMat L ) ) = ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat L ) j ) ) ) )
29	3, 17, 18	3jca 1242	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( R e. Ring /\ M e. B /\ L e. NN0 ) )
30	29	3ad2ant1 1082	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( R e. Ring /\ M e. B /\ L e. NN0 ) )
31		3simpc 1060	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( i e. N /\ j e. N ) )
32	9, 10, 11	decpmate 20571	. . . . 5 |- ( ( ( R e. Ring /\ M e. B /\ L e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( i ( M decompPMat L ) j ) = ( (coe1 ` ( i M j ) ) ` L ) )
33	30, 31, 32	syl2anc 693	. . . 4 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( i ( M decompPMat L ) j ) = ( (coe1 ` ( i M j ) ) ` L ) )
34	33	fveq2d 6195	. . 3 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( (algSc ` P ) ` ( i ( M decompPMat L ) j ) ) = ( (algSc ` P ) ` ( (coe1 ` ( i M j ) ) ` L ) ) )
35	34	mpt2eq3dva 6719	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( i e. N , j e. N |-> ( (algSc ` P ) ` ( i ( M decompPMat L ) j ) ) ) = ( i e. N , j e. N |-> ( (algSc ` P ) ` ( (coe1 ` ( i M j ) ) ` L ) ) ) )
36		simp1lr 1125	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> R e. Ring )
37		simp2 1062	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> i e. N )
38		simp3 1063	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> j e. N )
39	17	3ad2ant1 1082	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> M e. B )
40	10, 12, 11, 37, 38, 39	matecld 20232	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( i M j ) e. E )
41	18	3ad2ant1 1082	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> L e. NN0 )
42		eqid 2622	. . . . . . 7 |- (coe1 ` ( i M j ) ) = (coe1 ` ( i M j ) )
43		pmatcollpwscmat.k	. . . . . . 7 |- K = ( Base ` R )
44	42, 12, 9, 43	coe1fvalcl 19582	. . . . . 6 |- ( ( ( i M j ) e. E /\ L e. NN0 ) -> ( (coe1 ` ( i M j ) ) ` L ) e. K )
45	40, 41, 44	syl2anc 693	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( (coe1 ` ( i M j ) ) ` L ) e. K )
46		eqid 2622	. . . . . 6 |- (var1 ` R ) = (var1 ` R )
47		eqid 2622	. . . . . 6 |- ( .s ` P ) = ( .s ` P )
48		eqid 2622	. . . . . 6 |- (mulGrp ` P ) = (mulGrp ` P )
49		eqid 2622	. . . . . 6 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
50	43, 9, 46, 47, 48, 49, 26	ply1scltm 19651	. . . . 5 |- ( ( R e. Ring /\ ( (coe1 ` ( i M j ) ) ` L ) e. K ) -> ( (algSc ` P ) ` ( (coe1 ` ( i M j ) ) ` L ) ) = ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
51	36, 45, 50	syl2anc 693	. . . 4 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( (algSc ` P ) ` ( (coe1 ` ( i M j ) ) ` L ) ) = ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
52	51	mpt2eq3dva 6719	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( i e. N , j e. N |-> ( (algSc ` P ) ` ( (coe1 ` ( i M j ) ) ` L ) ) ) = ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) )
53		pmatcollpwscmat.e1	. . . . . . 7 |- .^ = (.g ` (mulGrp ` P ) )
54		pmatcollpwscmat.x	. . . . . . 7 |- X = (var1 ` R )
55		pmatcollpwscmat.u	. . . . . . 7 |- U = (algSc ` P )
56		pmatcollpwscmat.s	. . . . . . 7 |- S = (algSc ` P )
57	9, 10, 11, 13, 53, 54, 25, 19, 20, 55, 43, 12, 56, 14, 8	pmatcollpwscmatlem1 20594	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = if ( a = b , ( U ` ( (coe1 ` Q ) ` L ) ) , ( 0g ` P ) ) )
58		eqidd 2623	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) )
59		oveq12 6659	. . . . . . . . . . 11 |- ( ( i = a /\ j = b ) -> ( i M j ) = ( a M b ) )
60	59	fveq2d 6195	. . . . . . . . . 10 |- ( ( i = a /\ j = b ) -> (coe1 ` ( i M j ) ) = (coe1 ` ( a M b ) ) )
61	60	fveq1d 6193	. . . . . . . . 9 |- ( ( i = a /\ j = b ) -> ( (coe1 ` ( i M j ) ) ` L ) = ( (coe1 ` ( a M b ) ) ` L ) )
62	61	oveq1d 6665	. . . . . . . 8 |- ( ( i = a /\ j = b ) -> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
63	62	adantl 482	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) /\ ( i = a /\ j = b ) ) -> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
64		simprl 794	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> a e. N )
65		simprr 796	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> b e. N )
66		ovexd 6680	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) e. _V )
67	58, 63, 64, 65, 66	ovmpt2d 6788	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( a ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) b ) = ( ( (coe1 ` ( a M b ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
68		simpll 790	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> N e. Fin )
69	9	ply1ring 19618	. . . . . . . . . 10 |- ( R e. Ring -> P e. Ring )
70	69	adantl 482	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> P e. Ring )
71	70	adantr 481	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> P e. Ring )
72		pm3.22 465	. . . . . . . . . . 11 |- ( ( L e. NN0 /\ Q e. E ) -> ( Q e. E /\ L e. NN0 ) )
73	72	adantl 482	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( Q e. E /\ L e. NN0 ) )
74		eqid 2622	. . . . . . . . . . 11 |- (coe1 ` Q ) = (coe1 ` Q )
75	74, 12, 9, 43	coe1fvalcl 19582	. . . . . . . . . 10 |- ( ( Q e. E /\ L e. NN0 ) -> ( (coe1 ` Q ) ` L ) e. K )
76	73, 75	syl 17	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( (coe1 ` Q ) ` L ) e. K )
77	9, 55, 43, 12	ply1sclcl 19656	. . . . . . . . 9 |- ( ( R e. Ring /\ ( (coe1 ` Q ) ` L ) e. K ) -> ( U ` ( (coe1 ` Q ) ` L ) ) e. E )
78	3, 76, 77	syl2anc 693	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( U ` ( (coe1 ` Q ) ` L ) ) e. E )
79	68, 71, 78	3jca 1242	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( N e. Fin /\ P e. Ring /\ ( U ` ( (coe1 ` Q ) ` L ) ) e. E ) )
80		eqid 2622	. . . . . . . 8 |- ( 0g ` P ) = ( 0g ` P )
81	10, 12, 80, 14, 13	scmatscmide 20313	. . . . . . 7 |- ( ( ( N e. Fin /\ P e. Ring /\ ( U ` ( (coe1 ` Q ) ` L ) ) e. E ) /\ ( a e. N /\ b e. N ) ) -> ( a ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) b ) = if ( a = b , ( U ` ( (coe1 ` Q ) ` L ) ) , ( 0g ` P ) ) )
82	79, 81	sylan 488	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( a ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) b ) = if ( a = b , ( U ` ( (coe1 ` Q ) ` L ) ) , ( 0g ` P ) ) )
83	57, 67, 82	3eqtr4d 2666	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ ( a e. N /\ b e. N ) ) -> ( a ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) b ) = ( a ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) b ) )
84	83	ralrimivva 2971	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> A. a e. N A. b e. N ( a ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) b ) = ( a ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) b ) )
85		0nn0 11307	. . . . . . . 8 |- 0 e. NN0
86	85	a1i 11	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> 0 e. NN0 )
87	43, 9, 46, 47, 48, 49, 12	ply1tmcl 19642	. . . . . . 7 |- ( ( R e. Ring /\ ( (coe1 ` ( i M j ) ) ` L ) e. K /\ 0 e. NN0 ) -> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) e. E )
88	36, 45, 86, 87	syl3anc 1326	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) /\ i e. N /\ j e. N ) -> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) e. E )
89	10, 12, 11, 68, 71, 88	matbas2d 20229	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) e. B )
90	9, 10, 11, 12, 13, 14	1pmatscmul 20507	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ ( U ` ( (coe1 ` Q ) ` L ) ) e. E ) -> ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) e. B )
91	68, 3, 78, 90	syl3anc 1326	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) e. B )
92	10, 11	eqmat 20230	. . . . 5 |- ( ( ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) e. B /\ ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) e. B ) -> ( ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) <-> A. a e. N A. b e. N ( a ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) b ) = ( a ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) b ) ) )
93	89, 91, 92	syl2anc 693	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) <-> A. a e. N A. b e. N ( a ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) b ) = ( a ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) b ) ) )
94	84, 93	mpbird 247	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( i e. N , j e. N |-> ( ( (coe1 ` ( i M j ) ) ` L ) ( .s ` P ) ( 0 (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) )
95	52, 94	eqtrd 2656	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( i e. N , j e. N |-> ( (algSc ` P ) ` ( (coe1 ` ( i M j ) ) ` L ) ) ) = ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) )
96	28, 35, 95	3eqtrd 2660	1 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( L e. NN0 /\ Q e. E ) ) -> ( T ` ( M decompPMat L ) ) = ( ( U ` ( (coe1 ` Q ) ` L ) ) .* .1. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ifcif 4086   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   0cc0 9936   NN0cn0 11292   Basecbs 15857   .scvsca 15945   0gc0g 16100  ._gcmg 17540  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   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-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  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-mamu 20190  df-mat 20214  df-mat2pmat 20512  df-decpmat 20568

This theorem is referenced by:  pmatcollpwscmat  20596