Theorem q1pval 23913

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

Hypotheses

Ref	Expression
q1pval.q	|- Q = (quot1p ` R )
q1pval.p	|- P = (Poly1 ` R )
q1pval.b	|- B = ( Base ` P )
q1pval.d	|- D = ( deg1 ` R )
q1pval.m	|- .- = ( -g ` P )
q1pval.t	|- .x. = ( .r ` P )

Assertion

Ref	Expression
q1pval	|- ( ( F e. B /\ G e. B ) -> ( F Q G ) = ( iota_ q e. B ( D ` ( F .- ( q .x. G ) ) ) < ( D ` G ) ) )

Distinct variable groups:   B, q   F, q   G, q   P, q   R, q

Allowed substitution hints:   D( q)   Q( q)   .x. ( q)   .- ( q)

Proof of Theorem q1pval

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

Step	Hyp	Ref	Expression
1		q1pval.p	. . . . 5 |- P = (Poly1 ` R )
2		q1pval.b	. . . . 5 |- B = ( Base ` P )
3	1, 2	elbasfv 15920	. . . 4 |- ( G e. B -> R e. _V )
4		q1pval.q	. . . . 5 |- Q = (quot1p ` R )
5		fveq2 6191	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
6	5, 1	syl6eqr 2674	. . . . . . . 8 |- ( r = R -> (Poly1 ` r ) = P )
7	6	csbeq1d 3540	. . . . . . 7 |- ( r = R -> [_ (Poly1 ` r ) / p ]_ [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) = [_ P / p ]_ [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) )
8		fvex 6201	. . . . . . . . . 10 |- (Poly1 ` R ) e. _V
9	1, 8	eqeltri 2697	. . . . . . . . 9 |- P e. _V
10	9	a1i 11	. . . . . . . 8 |- ( r = R -> P e. _V )
11		fveq2 6191	. . . . . . . . . . . 12 |- ( p = P -> ( Base ` p ) = ( Base ` P ) )
12	11	adantl 482	. . . . . . . . . . 11 |- ( ( r = R /\ p = P ) -> ( Base ` p ) = ( Base ` P ) )
13	12, 2	syl6eqr 2674	. . . . . . . . . 10 |- ( ( r = R /\ p = P ) -> ( Base ` p ) = B )
14	13	csbeq1d 3540	. . . . . . . . 9 |- ( ( r = R /\ p = P ) -> [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) = [_ B / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) )
15		fvex 6201	. . . . . . . . . . . 12 |- ( Base ` P ) e. _V
16	2, 15	eqeltri 2697	. . . . . . . . . . 11 |- B e. _V
17	16	a1i 11	. . . . . . . . . 10 |- ( ( r = R /\ p = P ) -> B e. _V )
18		simpr 477	. . . . . . . . . . 11 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> b = B )
19		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
20	19	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( deg1 ` r ) = ( deg1 ` R ) )
21		q1pval.d	. . . . . . . . . . . . . . 15 |- D = ( deg1 ` R )
22	20, 21	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( deg1 ` r ) = D )
23		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( p = P -> ( -g ` p ) = ( -g ` P ) )
24	23	ad2antlr 763	. . . . . . . . . . . . . . . 16 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( -g ` p ) = ( -g ` P ) )
25		q1pval.m	. . . . . . . . . . . . . . . 16 |- .- = ( -g ` P )
26	24, 25	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( -g ` p ) = .- )
27		eqidd 2623	. . . . . . . . . . . . . . 15 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> f = f )
28		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( p = P -> ( .r ` p ) = ( .r ` P ) )
29	28	ad2antlr 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( .r ` p ) = ( .r ` P ) )
30		q1pval.t	. . . . . . . . . . . . . . . . 17 |- .x. = ( .r ` P )
31	29, 30	syl6eqr 2674	. . . . . . . . . . . . . . . 16 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( .r ` p ) = .x. )
32	31	oveqd 6667	. . . . . . . . . . . . . . 15 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( q ( .r ` p ) g ) = ( q .x. g ) )
33	26, 27, 32	oveq123d 6671	. . . . . . . . . . . . . 14 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( f ( -g ` p ) ( q ( .r ` p ) g ) ) = ( f .- ( q .x. g ) ) )
34	22, 33	fveq12d 6197	. . . . . . . . . . . . 13 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) = ( D ` ( f .- ( q .x. g ) ) ) )
35	22	fveq1d 6193	. . . . . . . . . . . . 13 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( ( deg1 ` r ) ` g ) = ( D ` g ) )
36	34, 35	breq12d 4666	. . . . . . . . . . . 12 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) <-> ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) )
37	18, 36	riotaeqbidv 6614	. . . . . . . . . . 11 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) = ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) )
38	18, 18, 37	mpt2eq123dv 6717	. . . . . . . . . 10 |- ( ( ( r = R /\ p = P ) /\ b = B ) -> ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
39	17, 38	csbied 3560	. . . . . . . . 9 |- ( ( r = R /\ p = P ) -> [_ B / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
40	14, 39	eqtrd 2656	. . . . . . . 8 |- ( ( r = R /\ p = P ) -> [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
41	10, 40	csbied 3560	. . . . . . 7 |- ( r = R -> [_ P / p ]_ [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
42	7, 41	eqtrd 2656	. . . . . 6 |- ( r = R -> [_ (Poly1 ` r ) / p ]_ [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
43		df-q1p 23892	. . . . . 6 |- quot1p = ( r e. _V |-> [_ (Poly1 ` r ) / p ]_ [_ ( Base ` p ) / b ]_ ( f e. b , g e. b |-> ( iota_ q e. b ( ( deg1 ` r ) ` ( f ( -g ` p ) ( q ( .r ` p ) g ) ) ) < ( ( deg1 ` r ) ` g ) ) ) )
44	16, 16	mpt2ex 7247	. . . . . 6 |- ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) e. _V
45	42, 43, 44	fvmpt 6282	. . . . 5 |- ( R e. _V -> (quot1p ` R ) = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
46	4, 45	syl5eq 2668	. . . 4 |- ( R e. _V -> Q = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
47	3, 46	syl 17	. . 3 |- ( G e. B -> Q = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
48	47	adantl 482	. 2 |- ( ( F e. B /\ G e. B ) -> Q = ( f e. B , g e. B |-> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) ) )
49		id 22	. . . . . . 7 |- ( f = F -> f = F )
50		oveq2 6658	. . . . . . 7 |- ( g = G -> ( q .x. g ) = ( q .x. G ) )
51	49, 50	oveqan12d 6669	. . . . . 6 |- ( ( f = F /\ g = G ) -> ( f .- ( q .x. g ) ) = ( F .- ( q .x. G ) ) )
52	51	fveq2d 6195	. . . . 5 |- ( ( f = F /\ g = G ) -> ( D ` ( f .- ( q .x. g ) ) ) = ( D ` ( F .- ( q .x. G ) ) ) )
53		fveq2 6191	. . . . . 6 |- ( g = G -> ( D ` g ) = ( D ` G ) )
54	53	adantl 482	. . . . 5 |- ( ( f = F /\ g = G ) -> ( D ` g ) = ( D ` G ) )
55	52, 54	breq12d 4666	. . . 4 |- ( ( f = F /\ g = G ) -> ( ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) <-> ( D ` ( F .- ( q .x. G ) ) ) < ( D ` G ) ) )
56	55	riotabidv 6613	. . 3 |- ( ( f = F /\ g = G ) -> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) = ( iota_ q e. B ( D ` ( F .- ( q .x. G ) ) ) < ( D ` G ) ) )
57	56	adantl 482	. 2 |- ( ( ( F e. B /\ G e. B ) /\ ( f = F /\ g = G ) ) -> ( iota_ q e. B ( D ` ( f .- ( q .x. g ) ) ) < ( D ` g ) ) = ( iota_ q e. B ( D ` ( F .- ( q .x. G ) ) ) < ( D ` G ) ) )
58		simpl 473	. 2 |- ( ( F e. B /\ G e. B ) -> F e. B )
59		simpr 477	. 2 |- ( ( F e. B /\ G e. B ) -> G e. B )
60		riotaex 6615	. . 3 |- ( iota_ q e. B ( D ` ( F .- ( q .x. G ) ) ) < ( D ` G ) ) e. _V
61	60	a1i 11	. 2 |- ( ( F e. B /\ G e. B ) -> ( iota_ q e. B ( D ` ( F .- ( q .x. G ) ) ) < ( D ` G ) ) e. _V )
62	48, 57, 58, 59, 61	ovmpt2d 6788	1 |- ( ( F e. B /\ G e. B ) -> ( F Q G ) = ( iota_ q e. B ( D ` ( F .- ( q .x. G ) ) ) < ( D ` G ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   class class class wbr 4653   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   < clt 10074   Basecbs 15857   .rcmulr 15942   -gcsg 17424  Poly₁cpl1 19547   deg₁ cdg1 23814  quot_1pcq1p 23887

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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-q1p 23892

This theorem is referenced by:  q1peqb  23914