Theorem cpmat 20514

Description: Value of the constructor of the set of all constant polynomial matrices, i.e. the set of all N x N matrices of polynomials over a ring R. (Contributed by AV, 15-Nov-2019.)

Hypotheses

Ref	Expression
cpmat.s	|- S = ( N ConstPolyMat R )
cpmat.p	|- P = (Poly1 ` R )
cpmat.c	|- C = ( N Mat P )
cpmat.b	|- B = ( Base ` C )

Assertion

Ref	Expression
cpmat	|- ( ( N e. Fin /\ R e. V ) -> S = { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } )

Distinct variable groups:   B, m   i, N, j, k, m   R, i, j, k, m

Allowed substitution hints:   B( i, j, k)   C( i, j, k, m)   P( i, j, k, m)   S( i, j, k, m)   V( i, j, k, m)

Proof of Theorem cpmat

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

Step	Hyp	Ref	Expression
1		cpmat.s	. 2 |- S = ( N ConstPolyMat R )
2		df-cpmat 20511	. . . 4 |- ConstPolyMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } )
3	2	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. V ) -> ConstPolyMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } ) )
4		simpl 473	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> n = N )
5		fveq2 6191	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
6	5	adantl 482	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> (Poly1 ` r ) = (Poly1 ` R ) )
7	4, 6	oveq12d 6668	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n Mat (Poly1 ` r ) ) = ( N Mat (Poly1 ` R ) ) )
8	7	fveq2d 6195	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat (Poly1 ` r ) ) ) = ( Base ` ( N Mat (Poly1 ` R ) ) ) )
9		cpmat.b	. . . . . . 7 |- B = ( Base ` C )
10		cpmat.c	. . . . . . . . 9 |- C = ( N Mat P )
11		cpmat.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
12	11	oveq2i 6661	. . . . . . . . 9 |- ( N Mat P ) = ( N Mat (Poly1 ` R ) )
13	10, 12	eqtri 2644	. . . . . . . 8 |- C = ( N Mat (Poly1 ` R ) )
14	13	fveq2i 6194	. . . . . . 7 |- ( Base ` C ) = ( Base ` ( N Mat (Poly1 ` R ) ) )
15	9, 14	eqtri 2644	. . . . . 6 |- B = ( Base ` ( N Mat (Poly1 ` R ) ) )
16	8, 15	syl6eqr 2674	. . . . 5 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat (Poly1 ` r ) ) ) = B )
17		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
18	17	adantl 482	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( 0g ` r ) = ( 0g ` R ) )
19	18	eqeq2d 2632	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) <-> ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) ) )
20	19	ralbidv 2986	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) <-> A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) ) )
21	4, 20	raleqbidv 3152	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( A. j e. n A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) <-> A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) ) )
22	4, 21	raleqbidv 3152	. . . . 5 |- ( ( n = N /\ r = R ) -> ( A. i e. n A. j e. n A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) <-> A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) ) )
23	16, 22	rabeqbidv 3195	. . . 4 |- ( ( n = N /\ r = R ) -> { m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } = { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } )
24	23	adantl 482	. . 3 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> { m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } = { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } )
25		simpl 473	. . 3 |- ( ( N e. Fin /\ R e. V ) -> N e. Fin )
26		elex 3212	. . . 4 |- ( R e. V -> R e. _V )
27	26	adantl 482	. . 3 |- ( ( N e. Fin /\ R e. V ) -> R e. _V )
28		fvex 6201	. . . . 5 |- ( Base ` C ) e. _V
29	9, 28	eqeltri 2697	. . . 4 |- B e. _V
30		rabexg 4812	. . . 4 |- ( B e. _V -> { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } e. _V )
31	29, 30	mp1i 13	. . 3 |- ( ( N e. Fin /\ R e. V ) -> { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } e. _V )
32	3, 24, 25, 27, 31	ovmpt2d 6788	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N ConstPolyMat R ) = { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } )
33	1, 32	syl5eq 2668	1 |- ( ( N e. Fin /\ R e. V ) -> S = { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   NNcn 11020   Basecbs 15857   0gc0g 16100  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   ConstPolyMat ccpmat 20508

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-cpmat 20511

This theorem is referenced by:  cpmatpmat  20515  cpmatel  20516  cpmatsubgpmat  20525