Theorem r1pval 23916

Description: Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
r1pval.e	|- E = (rem1p ` R )
r1pval.p	|- P = (Poly1 ` R )
r1pval.b	|- B = ( Base ` P )
r1pval.q	|- Q = (quot1p ` R )
r1pval.t	|- .x. = ( .r ` P )
r1pval.m	|- .- = ( -g ` P )

Assertion

Ref	Expression
r1pval	|- ( ( F e. B /\ G e. B ) -> ( F E G ) = ( F .- ( ( F Q G ) .x. G ) ) )

Proof of Theorem r1pval

Dummy variables b f g r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1pval.p	. . . . 5 |- P = (Poly1 ` R )
2		r1pval.b	. . . . 5 |- B = ( Base ` P )
3	1, 2	elbasfv 15920	. . . 4 |- ( F e. B -> R e. _V )
4	3	adantr 481	. . 3 |- ( ( F e. B /\ G e. B ) -> R e. _V )
5		r1pval.e	. . . 4 |- E = (rem1p ` R )
6		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
7	6, 1	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = P )
8	7	fveq2d 6195	. . . . . . . 8 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = ( Base ` P ) )
9	8, 2	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = B )
10	9	csbeq1d 3540	. . . . . 6 |- ( r = R -> [_ ( Base ` (Poly1 ` r ) ) / b ]_ ( f e. b , g e. b |-> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) ) = [_ B / b ]_ ( f e. b , g e. b |-> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) ) )
11		fvex 6201	. . . . . . . . 9 |- ( Base ` P ) e. _V
12	2, 11	eqeltri 2697	. . . . . . . 8 |- B e. _V
13	12	a1i 11	. . . . . . 7 |- ( r = R -> B e. _V )
14		simpr 477	. . . . . . . 8 |- ( ( r = R /\ b = B ) -> b = B )
15	7	fveq2d 6195	. . . . . . . . . . 11 |- ( r = R -> ( -g ` (Poly1 ` r ) ) = ( -g ` P ) )
16		r1pval.m	. . . . . . . . . . 11 |- .- = ( -g ` P )
17	15, 16	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> ( -g ` (Poly1 ` r ) ) = .- )
18		eqidd 2623	. . . . . . . . . 10 |- ( r = R -> f = f )
19	7	fveq2d 6195	. . . . . . . . . . . 12 |- ( r = R -> ( .r ` (Poly1 ` r ) ) = ( .r ` P ) )
20		r1pval.t	. . . . . . . . . . . 12 |- .x. = ( .r ` P )
21	19, 20	syl6eqr 2674	. . . . . . . . . . 11 |- ( r = R -> ( .r ` (Poly1 ` r ) ) = .x. )
22		fveq2 6191	. . . . . . . . . . . . 13 |- ( r = R -> (quot1p ` r ) = (quot1p ` R ) )
23		r1pval.q	. . . . . . . . . . . . 13 |- Q = (quot1p ` R )
24	22, 23	syl6eqr 2674	. . . . . . . . . . . 12 |- ( r = R -> (quot1p ` r ) = Q )
25	24	oveqd 6667	. . . . . . . . . . 11 |- ( r = R -> ( f (quot1p ` r ) g ) = ( f Q g ) )
26		eqidd 2623	. . . . . . . . . . 11 |- ( r = R -> g = g )
27	21, 25, 26	oveq123d 6671	. . . . . . . . . 10 |- ( r = R -> ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) = ( ( f Q g ) .x. g ) )
28	17, 18, 27	oveq123d 6671	. . . . . . . . 9 |- ( r = R -> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) = ( f .- ( ( f Q g ) .x. g ) ) )
29	28	adantr 481	. . . . . . . 8 |- ( ( r = R /\ b = B ) -> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) = ( f .- ( ( f Q g ) .x. g ) ) )
30	14, 14, 29	mpt2eq123dv 6717	. . . . . . 7 |- ( ( r = R /\ b = B ) -> ( f e. b , g e. b |-> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) ) = ( f e. B , g e. B |-> ( f .- ( ( f Q g ) .x. g ) ) ) )
31	13, 30	csbied 3560	. . . . . 6 |- ( r = R -> [_ B / b ]_ ( f e. b , g e. b |-> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) ) = ( f e. B , g e. B |-> ( f .- ( ( f Q g ) .x. g ) ) ) )
32	10, 31	eqtrd 2656	. . . . 5 |- ( r = R -> [_ ( Base ` (Poly1 ` r ) ) / b ]_ ( f e. b , g e. b |-> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) ) = ( f e. B , g e. B |-> ( f .- ( ( f Q g ) .x. g ) ) ) )
33		df-r1p 23893	. . . . 5 |- rem1p = ( r e. _V |-> [_ ( Base ` (Poly1 ` r ) ) / b ]_ ( f e. b , g e. b |-> ( f ( -g ` (Poly1 ` r ) ) ( ( f (quot1p ` r ) g ) ( .r ` (Poly1 ` r ) ) g ) ) ) )
34	12, 12	mpt2ex 7247	. . . . 5 |- ( f e. B , g e. B |-> ( f .- ( ( f Q g ) .x. g ) ) ) e. _V
35	32, 33, 34	fvmpt 6282	. . . 4 |- ( R e. _V -> (rem1p ` R ) = ( f e. B , g e. B |-> ( f .- ( ( f Q g ) .x. g ) ) ) )
36	5, 35	syl5eq 2668	. . 3 |- ( R e. _V -> E = ( f e. B , g e. B |-> ( f .- ( ( f Q g ) .x. g ) ) ) )
37	4, 36	syl 17	. 2 |- ( ( F e. B /\ G e. B ) -> E = ( f e. B , g e. B |-> ( f .- ( ( f Q g ) .x. g ) ) ) )
38		simpl 473	. . . 4 |- ( ( f = F /\ g = G ) -> f = F )
39		oveq12 6659	. . . . 5 |- ( ( f = F /\ g = G ) -> ( f Q g ) = ( F Q G ) )
40		simpr 477	. . . . 5 |- ( ( f = F /\ g = G ) -> g = G )
41	39, 40	oveq12d 6668	. . . 4 |- ( ( f = F /\ g = G ) -> ( ( f Q g ) .x. g ) = ( ( F Q G ) .x. G ) )
42	38, 41	oveq12d 6668	. . 3 |- ( ( f = F /\ g = G ) -> ( f .- ( ( f Q g ) .x. g ) ) = ( F .- ( ( F Q G ) .x. G ) ) )
43	42	adantl 482	. 2 |- ( ( ( F e. B /\ G e. B ) /\ ( f = F /\ g = G ) ) -> ( f .- ( ( f Q g ) .x. g ) ) = ( F .- ( ( F Q G ) .x. G ) ) )
44		simpl 473	. 2 |- ( ( F e. B /\ G e. B ) -> F e. B )
45		simpr 477	. 2 |- ( ( F e. B /\ G e. B ) -> G e. B )
46		ovexd 6680	. 2 |- ( ( F e. B /\ G e. B ) -> ( F .- ( ( F Q G ) .x. G ) ) e. _V )
47	37, 43, 44, 45, 46	ovmpt2d 6788	1 |- ( ( F e. B /\ G e. B ) -> ( F E G ) = ( F .- ( ( F Q G ) .x. G ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   .rcmulr 15942   -gcsg 17424  Poly₁cpl1 19547  quot_1pcq1p 23887  rem_1pcr1p 23888

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-slot 15861  df-base 15863  df-r1p 23893

This theorem is referenced by:  r1pcl  23917  r1pdeglt  23918  r1pid  23919  dvdsr1p  23921  ig1pdvds  23936