Theorem decpmatval0 20569

Description: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power, most general version. (Contributed by AV, 2-Dec-2019.)

Assertion

Ref	Expression
decpmatval0	|- ( ( M e. V /\ K e. NN0 ) -> ( M decompPMat K ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` K ) ) )

Distinct variable groups:   i, K, j   i, M, j

Allowed substitution hints:   V( i, j)

Proof of Theorem decpmatval0

Dummy variables k m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-decpmat 20568	. . 3 |- decompPMat = ( m e. _V , k e. NN0 |-> ( i e. dom dom m , j e. dom dom m |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
2	1	a1i 11	. 2 |- ( ( M e. V /\ K e. NN0 ) -> decompPMat = ( m e. _V , k e. NN0 |-> ( i e. dom dom m , j e. dom dom m |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) )
3		dmeq 5324	. . . . . 6 |- ( m = M -> dom m = dom M )
4	3	adantr 481	. . . . 5 |- ( ( m = M /\ k = K ) -> dom m = dom M )
5	4	dmeqd 5326	. . . 4 |- ( ( m = M /\ k = K ) -> dom dom m = dom dom M )
6		oveq 6656	. . . . . . 7 |- ( m = M -> ( i m j ) = ( i M j ) )
7	6	fveq2d 6195	. . . . . 6 |- ( m = M -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
8	7	adantr 481	. . . . 5 |- ( ( m = M /\ k = K ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
9		simpr 477	. . . . 5 |- ( ( m = M /\ k = K ) -> k = K )
10	8, 9	fveq12d 6197	. . . 4 |- ( ( m = M /\ k = K ) -> ( (coe1 ` ( i m j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` K ) )
11	5, 5, 10	mpt2eq123dv 6717	. . 3 |- ( ( m = M /\ k = K ) -> ( i e. dom dom m , j e. dom dom m |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` K ) ) )
12	11	adantl 482	. 2 |- ( ( ( M e. V /\ K e. NN0 ) /\ ( m = M /\ k = K ) ) -> ( i e. dom dom m , j e. dom dom m |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` K ) ) )
13		elex 3212	. . 3 |- ( M e. V -> M e. _V )
14	13	adantr 481	. 2 |- ( ( M e. V /\ K e. NN0 ) -> M e. _V )
15		simpr 477	. 2 |- ( ( M e. V /\ K e. NN0 ) -> K e. NN0 )
16		dmexg 7097	. . . . . 6 |- ( M e. V -> dom M e. _V )
17		dmexg 7097	. . . . . 6 |- ( dom M e. _V -> dom dom M e. _V )
18	16, 17	syl 17	. . . . 5 |- ( M e. V -> dom dom M e. _V )
19	18, 18	jca 554	. . . 4 |- ( M e. V -> ( dom dom M e. _V /\ dom dom M e. _V ) )
20	19	adantr 481	. . 3 |- ( ( M e. V /\ K e. NN0 ) -> ( dom dom M e. _V /\ dom dom M e. _V ) )
21		mpt2exga 7246	. . 3 |- ( ( dom dom M e. _V /\ dom dom M e. _V ) -> ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` K ) ) e. _V )
22	20, 21	syl 17	. 2 |- ( ( M e. V /\ K e. NN0 ) -> ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` K ) ) e. _V )
23	2, 12, 14, 15, 22	ovmpt2d 6788	1 |- ( ( M e. V /\ K e. NN0 ) -> ( M decompPMat K ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   NN0cn0 11292  coe₁cco1 19548   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

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-decpmat 20568

This theorem is referenced by:  decpmatval  20570