Theorem cpmatel 20516

Description: Property of a constant polynomial matrix. (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
cpmatel	|- ( ( N e. Fin /\ R e. V /\ M e. B ) -> ( M e. S <-> A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) )

Distinct variable groups:   i, N, j, k   R, i, j, k   i, M, j, k

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

Proof of Theorem cpmatel

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmat.s	. . . . . 6 |- S = ( N ConstPolyMat R )
2		cpmat.p	. . . . . 6 |- P = (Poly1 ` R )
3		cpmat.c	. . . . . 6 |- C = ( N Mat P )
4		cpmat.b	. . . . . 6 |- B = ( Base ` C )
5	1, 2, 3, 4	cpmat 20514	. . . . 5 |- ( ( 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 ) } )
6	5	3adant3 1081	. . . 4 |- ( ( N e. Fin /\ R e. V /\ M e. B ) -> S = { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } )
7	6	eleq2d 2687	. . 3 |- ( ( N e. Fin /\ R e. V /\ M e. B ) -> ( M e. S <-> M e. { m e. B | A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) } ) )
8		oveq 6656	. . . . . . . . 9 |- ( m = M -> ( i m j ) = ( i M j ) )
9	8	fveq2d 6195	. . . . . . . 8 |- ( m = M -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
10	9	fveq1d 6193	. . . . . . 7 |- ( m = M -> ( (coe1 ` ( i m j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` k ) )
11	10	eqeq1d 2624	. . . . . 6 |- ( m = M -> ( ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) <-> ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) )
12	11	ralbidv 2986	. . . . 5 |- ( m = M -> ( A. k e. NN ( (coe1 ` ( i m j ) ) ` k ) = ( 0g ` R ) <-> A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) )
13	12	2ralbidv 2989	. . . 4 |- ( m = M -> ( 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 ) ) )
14	13	elrab 3363	. . 3 |- ( M e. { m e. B | 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 ) ) )
15	7, 14	syl6bb 276	. 2 |- ( ( N e. Fin /\ R e. V /\ M e. B ) -> ( M e. S <-> ( M e. B /\ A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) ) )
16	15	3anibar 1229	1 |- ( ( N e. Fin /\ R e. V /\ M e. B ) -> ( M e. S <-> A. i e. N A. j e. N A. k e. NN ( (coe1 ` ( i M j ) ) ` k ) = ( 0g ` R ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   ` cfv 5888  (class class class)co 6650   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:  cpmatelimp  20517  cpmatel2  20518  1elcpmat  20520  cpmatmcl  20524  m2cpm  20546