Theorem pmattomply1f1lem 31257

Description: Lemma for pmattomply1f1 31269. (Contributed by AV, 14-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 )

Assertion

Ref	Expression
pmattomply1f1lem	|- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( U F k ) .* ( k .^ X ) ) ) ) ) ` K ) = ( U F K ) )

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

Allowed substitution hints:   A( m)   P( i, j, m)   Q( i, j, m)   R( m)   .^ ( i, j, m)   F( m)   .* ( i, j, m)   K( i, j, m)   L( i, j, m)   X( i, j, k, m)

Proof of Theorem pmattomply1f1lem

Dummy variables x b y l are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmattomply1.q	. . 3 |- Q = (Poly1 ` A )
2		pmattomply1.l	. . 3 |- L = ( Base ` Q )
3		pmattomply1.x	. . 3 |- X = (var1 ` A )
4		pmattomply1.e	. . 3 |- .^ = (.g ` (mulGrp ` Q ) )
5		pmattomply1.a	. . . . 5 |- A = ( N Mat R )
6	5	matrng 18442	. . . 4 |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring )
7	6	adantr 465	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> A e. Ring )
8		eqid 2451	. . 3 |- ( Base ` A ) = ( Base ` A )
9		pmattomply1.m	. . 3 |- .* = ( .s ` Q )
10		eqid 2451	. . 3 |- ( 0g ` A ) = ( 0g ` A )
11		simpll 753	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) /\ k e. NN0 ) -> ( N e. Fin /\ R e. Ring ) )
12		simprl 755	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> U e. B )
13	12	anim1i 568	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) /\ k e. NN0 ) -> ( U e. B /\ k e. NN0 ) )
14		pmattomply1.p	. . . . . 6 |- P = (Poly1 ` R )
15		pmattomply1.c	. . . . . 6 |- C = ( N Mat P )
16		pmattomply1.b	. . . . . 6 |- B = ( Base ` C )
17		pmattomply1.f	. . . . . . 7 |- F = ( m e. B , k e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
18		oveq 6198	. . . . . . . . . . . 12 |- ( m = b -> ( i m j ) = ( i b j ) )
19	18	fveq2d 5795	. . . . . . . . . . 11 |- ( m = b -> (coe1 ` ( i m j ) ) = (coe1 ` ( i b j ) ) )
20	19	fveq1d 5793	. . . . . . . . . 10 |- ( m = b -> ( (coe1 ` ( i m j ) ) ` k ) = ( (coe1 ` ( i b j ) ) ` k ) )
21	20	mpt2eq3dv 6253	. . . . . . . . 9 |- ( m = b -> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i b j ) ) ` k ) ) )
22		fveq2 5791	. . . . . . . . . 10 |- ( k = l -> ( (coe1 ` ( i b j ) ) ` k ) = ( (coe1 ` ( i b j ) ) ` l ) )
23	22	mpt2eq3dv 6253	. . . . . . . . 9 |- ( k = l -> ( i e. N , j e. N |-> ( (coe1 ` ( i b j ) ) ` k ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i b j ) ) ` l ) ) )
24	21, 23	cbvmpt2v 6267	. . . . . . . 8 |- ( m e. B , k e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) = ( b e. B , l e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i b j ) ) ` l ) ) )
25		oveq 6198	. . . . . . . . . . . 12 |- ( b = y -> ( i b j ) = ( i y j ) )
26	25	fveq2d 5795	. . . . . . . . . . 11 |- ( b = y -> (coe1 ` ( i b j ) ) = (coe1 ` ( i y j ) ) )
27	26	fveq1d 5793	. . . . . . . . . 10 |- ( b = y -> ( (coe1 ` ( i b j ) ) ` l ) = ( (coe1 ` ( i y j ) ) ` l ) )
28	27	mpt2eq3dv 6253	. . . . . . . . 9 |- ( b = y -> ( i e. N , j e. N |-> ( (coe1 ` ( i b j ) ) ` l ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i y j ) ) ` l ) ) )
29		fveq2 5791	. . . . . . . . . 10 |- ( l = x -> ( (coe1 ` ( i y j ) ) ` l ) = ( (coe1 ` ( i y j ) ) ` x ) )
30	29	mpt2eq3dv 6253	. . . . . . . . 9 |- ( l = x -> ( i e. N , j e. N |-> ( (coe1 ` ( i y j ) ) ` l ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i y j ) ) ` x ) ) )
31	28, 30	cbvmpt2v 6267	. . . . . . . 8 |- ( b e. B , l e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i b j ) ) ` l ) ) ) = ( y e. B , x e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i y j ) ) ` x ) ) )
32	24, 31	eqtri 2480	. . . . . . 7 |- ( m e. B , k e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) = ( y e. B , x e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i y j ) ) ` x ) ) )
33	17, 32	eqtri 2480	. . . . . 6 |- F = ( y e. B , x e. NN0 |-> ( i e. N , j e. N |-> ( (coe1 ` ( i y j ) ) ` x ) ) )
34	14, 15, 16, 33, 5, 8	pmatcollpw1lem2 31226	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ k e. NN0 ) ) -> ( U F k ) e. ( Base ` A ) )
35	11, 13, 34	syl2anc 661	. . . 4 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) /\ k e. NN0 ) -> ( U F k ) e. ( Base ` A ) )
36	35	ralrimiva 2822	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> A. k e. NN0 ( U F k ) e. ( Base ` A ) )
37		simpll 753	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> N e. Fin )
38		simpr 461	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> R e. Ring )
39	38	adantr 465	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> R e. Ring )
40	14, 15, 16, 17, 5	pmatcollpwfsupp 31231	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ U e. B ) -> ( k e. NN0 |-> ( U F k ) ) finSupp ( 0g ` A ) )
41	37, 39, 12, 40	syl3anc 1219	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> ( k e. NN0 |-> ( U F k ) ) finSupp ( 0g ` A ) )
42		simprr 756	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> K e. NN0 )
43	1, 2, 3, 4, 7, 8, 9, 10, 36, 41, 42	gsummoncoe1 30987	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( U F k ) .* ( k .^ X ) ) ) ) ) ` K ) = [_ K / k ]_ ( U F k ) )
44		csbov2g 6228	. . 3 |- ( K e. NN0 -> [_ K / k ]_ ( U F k ) = ( U F [_ K / k ]_ k ) )
45	44	ad2antll 728	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> [_ K / k ]_ ( U F k ) = ( U F [_ K / k ]_ k ) )
46		csbvarg 3800	. . . 4 |- ( K e. NN0 -> [_ K / k ]_ k = K )
47	46	ad2antll 728	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> [_ K / k ]_ k = K )
48	47	oveq2d 6208	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> ( U F [_ K / k ]_ k ) = ( U F K ) )
49	43, 45, 48	3eqtrd 2496	1 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( U e. B /\ K e. NN0 ) ) -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( U F k ) .* ( k .^ X ) ) ) ) ) ` K ) = ( U F K ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   [_csb 3388   class class class wbr 4392   |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   |-> cmpt2 6194   Fincfn 7412   finSupp cfsupp 7723   NN0cn0 10682   Basecbs 14278   .scvsca 14346   0gc0g 14482   gsum_g cgsu 14483  ._gcmg 15518  mulGrpcmgp 16698   Ringcrg 16753  var₁cv1 17741  Poly₁cpl1 17742  coe₁cco1 17743   Mat cmat 18391

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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

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 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-ot 3986  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-ofr 6423  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-fz 11541  df-fzo 11652  df-seq 11910  df-hash 12207  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-ip 14360  df-tset 14361  df-ple 14362  df-ds 14364  df-hom 14366  df-cco 14367  df-0g 14484  df-gsum 14485  df-prds 14490  df-pws 14492  df-mre 14628  df-mrc 14629  df-acs 14631  df-mnd 15519  df-mhm 15568  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-mulg 15652  df-subg 15782  df-ghm 15849  df-cntz 15939  df-cmn 16385  df-abl 16386  df-mgp 16699  df-ur 16711  df-rng 16755  df-subrg 16971  df-lmod 17058  df-lss 17122  df-sra 17361  df-rgmod 17362  df-psr 17531  df-mvr 17532  df-mpl 17533  df-opsr 17535  df-psr1 17745  df-vr1 17746  df-ply1 17747  df-coe1 17748  df-dsmm 18268  df-frlm 18283  df-mamu 18392  df-mat 18393

This theorem is referenced by:  pmattomply1f1  31269