Theorem pmattomply1fo 31268

Description: The transformation of matrices over polynomials into polynomials over matrices is a function mapping matrices over polynomials onto polynomials over matrices. (Contributed by AV, 12-Oct-2019.)

Hypotheses

Ref	Expression
pmattomply1.p	|- P = (Poly1 ` R )
pmattomply1.c	|- C = ( N Mat P )
pmattomply1.b	|- B = ( Base ` C )
pmattomply1.f	|- F = ( m e. B , k e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
pmattomply1.m	|- .* = ( .s ` Q )
pmattomply1.e	|- .^ = (.g ` (mulGrp ` Q ) )
pmattomply1.x	|- X = (var1 ` A )
pmattomply1.a	|- A = ( N Mat R )
pmattomply1.q	|- Q = (Poly1 ` A )
pmattomply1.l	|- L = ( Base ` Q )
pmattomply1.t	|- T = ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m F k ) .* ( k .^ X ) ) ) ) )

Assertion

Ref	Expression
pmattomply1fo	|- ( ( N e. Fin /\ R e. Ring ) -> T : B -onto-> L )

Distinct variable groups:   B, k, m, i, j   i, N, j, k   R, i, j, k   m, F   Q, m   m, X   .* , m   .^ , m   A, i, j, k   C, i, j, k, m   i, F, j, k   k, L   m, N   P, k   .* , k   R, m   Q, k   .^ , k   i, L, j, m   P, i, j, m   k, X

Allowed substitution hints:   A( m)   Q( i, j)   T( i, j, k, m)   .^ ( i, j)   .* ( i, j)   X( i, j)

Proof of Theorem pmattomply1fo

Dummy variables l u w x y z f p v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmattomply1.p	. . 3 |- P = (Poly1 ` R )
2		pmattomply1.c	. . 3 |- C = ( N Mat P )
3		pmattomply1.b	. . 3 |- B = ( Base ` C )
4		pmattomply1.f	. . 3 |- F = ( m e. B , k e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
5		pmattomply1.m	. . 3 |- .* = ( .s ` Q )
6		pmattomply1.e	. . 3 |- .^ = (.g ` (mulGrp ` Q ) )
7		pmattomply1.x	. . 3 |- X = (var1 ` A )
8		pmattomply1.a	. . 3 |- A = ( N Mat R )
9		pmattomply1.q	. . 3 |- Q = (Poly1 ` A )
10		pmattomply1.l	. . 3 |- L = ( Base ` Q )
11		pmattomply1.t	. . 3 |- T = ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m F k ) .* ( k .^ X ) ) ) ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	pmattomply1f 31258	. 2 |- ( ( N e. Fin /\ R e. Ring ) -> T : B --> L )
13		oveq 6196	. . . . . . . . . . 11 |- ( m = w -> ( i m j ) = ( i w j ) )
14	13	fveq2d 5793	. . . . . . . . . 10 |- ( m = w -> (coe1 ` ( i m j ) ) = (coe1 ` ( i w j ) ) )
15	14	fveq1d 5791	. . . . . . . . 9 |- ( m = w -> ( (coe1 ` ( i m j ) ) ` k ) = ( (coe1 ` ( i w j ) ) ` k ) )
16	15	mpt2eq3dv 6251	. . . . . . . 8 |- ( m = w -> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i w j ) ) ` k ) ) )
17		fveq2 5789	. . . . . . . . 9 |- ( k = u -> ( (coe1 ` ( i w j ) ) ` k ) = ( (coe1 ` ( i w j ) ) ` u ) )
18	17	mpt2eq3dv 6251	. . . . . . . 8 |- ( k = u -> ( i e. N , j e. N |-> ( (coe1 ` ( i w j ) ) ` k ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i w j ) ) ` u ) ) )
19	16, 18	cbvmpt2v 6265	. . . . . . 7 |- ( m e. B , k e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) = ( w e. B , u e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i w j ) ) ` u ) ) )
20		oveq 6196	. . . . . . . . . . 11 |- ( w = m -> ( i w j ) = ( i m j ) )
21	20	fveq2d 5793	. . . . . . . . . 10 |- ( w = m -> (coe1 ` ( i w j ) ) = (coe1 ` ( i m j ) ) )
22	21	fveq1d 5791	. . . . . . . . 9 |- ( w = m -> ( (coe1 ` ( i w j ) ) ` u ) = ( (coe1 ` ( i m j ) ) ` u ) )
23	22	mpt2eq3dv 6251	. . . . . . . 8 |- ( w = m -> ( i e. N , j e. N |-> ( (coe1 ` ( i w j ) ) ` u ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` u ) ) )
24		fveq2 5789	. . . . . . . . . 10 |- ( u = z -> ( (coe1 ` ( i m j ) ) ` u ) = ( (coe1 ` ( i m j ) ) ` z ) )
25	24	mpt2eq3dv 6251	. . . . . . . . 9 |- ( u = z -> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` u ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` z ) ) )
26		oveq1 6197	. . . . . . . . . . . 12 |- ( i = x -> ( i m j ) = ( x m j ) )
27	26	fveq2d 5793	. . . . . . . . . . 11 |- ( i = x -> (coe1 ` ( i m j ) ) = (coe1 ` ( x m j ) ) )
28	27	fveq1d 5791	. . . . . . . . . 10 |- ( i = x -> ( (coe1 ` ( i m j ) ) ` z ) = ( (coe1 ` ( x m j ) ) ` z ) )
29		oveq2 6198	. . . . . . . . . . . 12 |- ( j = y -> ( x m j ) = ( x m y ) )
30	29	fveq2d 5793	. . . . . . . . . . 11 |- ( j = y -> (coe1 ` ( x m j ) ) = (coe1 ` ( x m y ) ) )
31	30	fveq1d 5791	. . . . . . . . . 10 |- ( j = y -> ( (coe1 ` ( x m j ) ) ` z ) = ( (coe1 ` ( x m y ) ) ` z ) )
32	28, 31	cbvmpt2v 6265	. . . . . . . . 9 |- ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` z ) ) = ( x e. N , y e. N |-> ( (coe1 ` ( x m y ) ) ` z ) )
33	25, 32	syl6eq 2508	. . . . . . . 8 |- ( u = z -> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` u ) ) = ( x e. N , y e. N |-> ( (coe1 ` ( x m y ) ) ` z ) ) )
34	23, 33	cbvmpt2v 6265	. . . . . . 7 |- ( w e. B , u e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i w j ) ) ` u ) ) ) = ( m e. B , z e. NN0 |-> ( x e. N , y e. N |-> ( (coe1 ` ( x m y ) ) ` z ) ) )
35	4, 19, 34	3eqtri 2484	. . . . . 6 |- F = ( m e. B , z e. NN0 |-> ( x e. N , y e. N |-> ( (coe1 ` ( x m y ) ) ` z ) ) )
36		oveq2 6198	. . . . . . . . . . 11 |- ( k = z -> ( m F k ) = ( m F z ) )
37		oveq1 6197	. . . . . . . . . . 11 |- ( k = z -> ( k .^ X ) = ( z .^ X ) )
38	36, 37	oveq12d 6208	. . . . . . . . . 10 |- ( k = z -> ( ( m F k ) .* ( k .^ X ) ) = ( ( m F z ) .* ( z .^ X ) ) )
39	38	cbvmptv 4481	. . . . . . . . 9 |- ( k e. NN0 |-> ( ( m F k ) .* ( k .^ X ) ) ) = ( z e. NN0 |-> ( ( m F z ) .* ( z .^ X ) ) )
40	39	oveq2i 6201	. . . . . . . 8 |- ( Q gsumg ( k e. NN0 |-> ( ( m F k ) .* ( k .^ X ) ) ) ) = ( Q gsumg ( z e. NN0 |-> ( ( m F z ) .* ( z .^ X ) ) ) )
41	40	mpteq2i 4473	. . . . . . 7 |- ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m F k ) .* ( k .^ X ) ) ) ) ) = ( m e. B |-> ( Q gsumg ( z e. NN0 |-> ( ( m F z ) .* ( z .^ X ) ) ) ) )
42	11, 41	eqtri 2480	. . . . . 6 |- T = ( m e. B |-> ( Q gsumg ( z e. NN0 |-> ( ( m F z ) .* ( z .^ X ) ) ) ) )
43		eqid 2451	. . . . . 6 |- ( .s ` P ) = ( .s ` P )
44		eqid 2451	. . . . . 6 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
45		eqid 2451	. . . . . 6 |- (var1 ` R ) = (var1 ` R )
46		fveq2 5789	. . . . . . . . . . . . . 14 |- ( l = v -> (coe1 ` l ) = (coe1 ` v ) )
47	46	fveq1d 5791	. . . . . . . . . . . . 13 |- ( l = v -> ( (coe1 ` l ) ` k ) = ( (coe1 ` v ) ` k ) )
48	47	oveqd 6207	. . . . . . . . . . . 12 |- ( l = v -> ( i ( (coe1 ` l ) ` k ) j ) = ( i ( (coe1 ` v ) ` k ) j ) )
49	48	oveq1d 6205	. . . . . . . . . . 11 |- ( l = v -> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
50	49	mpteq2dv 4477	. . . . . . . . . 10 |- ( l = v -> ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( k e. NN0 |-> ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) )
51	50	oveq2d 6206	. . . . . . . . 9 |- ( l = v -> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) = ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) )
52	51	mpt2eq3dv 6251	. . . . . . . 8 |- ( l = v -> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) = ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) )
53		oveq1 6197	. . . . . . . . . . . 12 |- ( i = x -> ( i ( (coe1 ` v ) ` k ) j ) = ( x ( (coe1 ` v ) ` k ) j ) )
54	53	oveq1d 6205	. . . . . . . . . . 11 |- ( i = x -> ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( x ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
55	54	mpteq2dv 4477	. . . . . . . . . 10 |- ( i = x -> ( k e. NN0 |-> ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( k e. NN0 |-> ( ( x ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) )
56	55	oveq2d 6206	. . . . . . . . 9 |- ( i = x -> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) = ( P gsumg ( k e. NN0 |-> ( ( x ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) )
57		oveq2 6198	. . . . . . . . . . . . 13 |- ( j = y -> ( x ( (coe1 ` v ) ` k ) j ) = ( x ( (coe1 ` v ) ` k ) y ) )
58	57	oveq1d 6205	. . . . . . . . . . . 12 |- ( j = y -> ( ( x ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( x ( (coe1 ` v ) ` k ) y ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
59	58	mpteq2dv 4477	. . . . . . . . . . 11 |- ( j = y -> ( k e. NN0 |-> ( ( x ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( k e. NN0 |-> ( ( x ( (coe1 ` v ) ` k ) y ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) )
60		fveq2 5789	. . . . . . . . . . . . . 14 |- ( k = z -> ( (coe1 ` v ) ` k ) = ( (coe1 ` v ) ` z ) )
61	60	oveqd 6207	. . . . . . . . . . . . 13 |- ( k = z -> ( x ( (coe1 ` v ) ` k ) y ) = ( x ( (coe1 ` v ) ` z ) y ) )
62		oveq1 6197	. . . . . . . . . . . . 13 |- ( k = z -> ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) = ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) )
63	61, 62	oveq12d 6208	. . . . . . . . . . . 12 |- ( k = z -> ( ( x ( (coe1 ` v ) ` k ) y ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) = ( ( x ( (coe1 ` v ) ` z ) y ) ( .s ` P ) ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
64	63	cbvmptv 4481	. . . . . . . . . . 11 |- ( k e. NN0 |-> ( ( x ( (coe1 ` v ) ` k ) y ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( z e. NN0 |-> ( ( x ( (coe1 ` v ) ` z ) y ) ( .s ` P ) ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) )
65	59, 64	syl6eq 2508	. . . . . . . . . 10 |- ( j = y -> ( k e. NN0 |-> ( ( x ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) = ( z e. NN0 |-> ( ( x ( (coe1 ` v ) ` z ) y ) ( .s ` P ) ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) )
66	65	oveq2d 6206	. . . . . . . . 9 |- ( j = y -> ( P gsumg ( k e. NN0 |-> ( ( x ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) = ( P gsumg ( z e. NN0 |-> ( ( x ( (coe1 ` v ) ` z ) y ) ( .s ` P ) ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) )
67	56, 66	cbvmpt2v 6265	. . . . . . . 8 |- ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` v ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) = ( x e. N , y e. N |-> ( P gsumg ( z e. NN0 |-> ( ( x ( (coe1 ` v ) ` z ) y ) ( .s ` P ) ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) )
68	52, 67	syl6eq 2508	. . . . . . 7 |- ( l = v -> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) = ( x e. N , y e. N |-> ( P gsumg ( z e. NN0 |-> ( ( x ( (coe1 ` v ) ` z ) y ) ( .s ` P ) ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) )
69	68	cbvmptv 4481	. . . . . 6 |- ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) = ( v e. L |-> ( x e. N , y e. N |-> ( P gsumg ( z e. NN0 |-> ( ( x ( (coe1 ` v ) ` z ) y ) ( .s ` P ) ( z (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) )
70	1, 2, 3, 35, 5, 6, 7, 8, 9, 10, 42, 43, 44, 45, 69	mp2pm2mp 31266	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ p e. L ) -> ( T ` ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) = p )
71	70	3expa 1188	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) -> ( T ` ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) = p )
72		eqid 2451	. . . . . . . . 9 |- ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) = ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) )
73	8, 9, 10, 1, 43, 44, 45, 72, 2, 3	mply1topmatcl 31252	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ p e. L ) -> ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) e. B )
74	73	3expa 1188	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) -> ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) e. B )
75		simpr 461	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) /\ f = ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) -> f = ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) )
76	75	fveq2d 5793	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) /\ f = ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) -> ( T ` f ) = ( T ` ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) )
77	76	eqeq2d 2465	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) /\ f = ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) -> ( p = ( T ` f ) <-> p = ( T ` ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) ) )
78	74, 77	rspcedv 3173	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) -> ( p = ( T ` ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) -> E. f e. B p = ( T ` f ) ) )
79	78	com12 31	. . . . 5 |- ( p = ( T ` ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) -> ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) -> E. f e. B p = ( T ` f ) ) )
80	79	eqcoms 2463	. . . 4 |- ( ( T ` ( ( l e. L |-> ( i e. N , j e. N |-> ( P gsumg ( k e. NN0 |-> ( ( i ( (coe1 ` l ) ` k ) j ) ( .s ` P ) ( k (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) ) ) ) ) ` p ) ) = p -> ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) -> E. f e. B p = ( T ` f ) ) )
81	71, 80	mpcom 36	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ p e. L ) -> E. f e. B p = ( T ` f ) )
82	81	ralrimiva 2822	. 2 |- ( ( N e. Fin /\ R e. Ring ) -> A. p e. L E. f e. B p = ( T ` f ) )
83		dffo3 5957	. 2 |- ( T : B -onto-> L <-> ( T : B --> L /\ A. p e. L E. f e. B p = ( T ` f ) ) )
84	12, 82, 83	sylanbrc 664	1 |- ( ( N e. Fin /\ R e. Ring ) -> T : B -onto-> L )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2795   E.wrex 2796   |-> cmpt 4448   -->wf 5512   -onto->wfo 5514   ` cfv 5516  (class class class)co 6190   |-> cmpt2 6192   Fincfn 7410   NN0cn0 10680   Basecbs 14276   .scvsca 14344   gsum_g cgsu 14481  ._gcmg 15516  mulGrpcmgp 16696   Ringcrg 16751  var₁cv1 17739  Poly₁cpl1 17740  coe₁cco1 17741   Mat cmat 18389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-ot 3984  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-ofr 6421  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-0g 14482  df-gsum 14483  df-prds 14488  df-pws 14490  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-ghm 15847  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-srg 16713  df-rng 16753  df-subrg 16969  df-lmod 17056  df-lss 17120  df-sra 17359  df-rgmod 17360  df-psr 17529  df-mvr 17530  df-mpl 17531  df-opsr 17533  df-psr1 17743  df-vr1 17744  df-ply1 17745  df-coe1 17746  df-dsmm 18266  df-frlm 18281  df-mamu 18390  df-mat 18391

This theorem is referenced by:  pmattomply1f1o  31269