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Theorem q1pval 23851
 Description: Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
q1pval.q 𝑄 = (quot1p𝑅)
q1pval.p 𝑃 = (Poly1𝑅)
q1pval.b 𝐵 = (Base‘𝑃)
q1pval.d 𝐷 = ( deg1𝑅)
q1pval.m = (-g𝑃)
q1pval.t · = (.r𝑃)
Assertion
Ref Expression
q1pval ((𝐹𝐵𝐺𝐵) → (𝐹𝑄𝐺) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
Distinct variable groups:   𝐵,𝑞   𝐹,𝑞   𝐺,𝑞   𝑃,𝑞   𝑅,𝑞
Allowed substitution hints:   𝐷(𝑞)   𝑄(𝑞)   · (𝑞)   (𝑞)

Proof of Theorem q1pval
Dummy variables 𝑏 𝑓 𝑔 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 q1pval.p . . . . 5 𝑃 = (Poly1𝑅)
2 q1pval.b . . . . 5 𝐵 = (Base‘𝑃)
31, 2elbasfv 15860 . . . 4 (𝐺𝐵𝑅 ∈ V)
4 q1pval.q . . . . 5 𝑄 = (quot1p𝑅)
5 fveq2 6158 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65, 1syl6eqr 2673 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
76csbeq1d 3526 . . . . . . 7 (𝑟 = 𝑅(Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))) = 𝑃 / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
8 fvex 6168 . . . . . . . . . 10 (Poly1𝑅) ∈ V
91, 8eqeltri 2694 . . . . . . . . 9 𝑃 ∈ V
109a1i 11 . . . . . . . 8 (𝑟 = 𝑅𝑃 ∈ V)
11 fveq2 6158 . . . . . . . . . . . 12 (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
1211adantl 482 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
1312, 2syl6eqr 2673 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
1413csbeq1d 3526 . . . . . . . . 9 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))) = 𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
15 fvex 6168 . . . . . . . . . . . 12 (Base‘𝑃) ∈ V
162, 15eqeltri 2694 . . . . . . . . . . 11 𝐵 ∈ V
1716a1i 11 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → 𝐵 ∈ V)
18 simpr 477 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
19 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
2019ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = ( deg1𝑅))
21 q1pval.d . . . . . . . . . . . . . . 15 𝐷 = ( deg1𝑅)
2220, 21syl6eqr 2673 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = 𝐷)
23 fveq2 6158 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑃 → (-g𝑝) = (-g𝑃))
2423ad2antlr 762 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = (-g𝑃))
25 q1pval.m . . . . . . . . . . . . . . . 16 = (-g𝑃)
2624, 25syl6eqr 2673 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = )
27 eqidd 2622 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
28 fveq2 6158 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑃 → (.r𝑝) = (.r𝑃))
2928ad2antlr 762 . . . . . . . . . . . . . . . . 17 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = (.r𝑃))
30 q1pval.t . . . . . . . . . . . . . . . . 17 · = (.r𝑃)
3129, 30syl6eqr 2673 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = · )
3231oveqd 6632 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r𝑝)𝑔) = (𝑞 · 𝑔))
3326, 27, 32oveq123d 6636 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g𝑝)(𝑞(.r𝑝)𝑔)) = (𝑓 (𝑞 · 𝑔)))
3422, 33fveq12d 6164 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) = (𝐷‘(𝑓 (𝑞 · 𝑔))))
3522fveq1d 6160 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
3634, 35breq12d 4636 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔) ↔ (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3718, 36riotaeqbidv 6579 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔)) = (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3818, 18, 37mpt2eq123dv 6682 . . . . . . . . . 10 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
3917, 38csbied 3546 . . . . . . . . 9 ((𝑟 = 𝑅𝑝 = 𝑃) → 𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
4014, 39eqtrd 2655 . . . . . . . 8 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
4110, 40csbied 3546 . . . . . . 7 (𝑟 = 𝑅𝑃 / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
427, 41eqtrd 2655 . . . . . 6 (𝑟 = 𝑅(Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
43 df-q1p 23830 . . . . . 6 quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
4416, 16mpt2ex 7207 . . . . . 6 (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))) ∈ V
4542, 43, 44fvmpt 6249 . . . . 5 (𝑅 ∈ V → (quot1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
464, 45syl5eq 2667 . . . 4 (𝑅 ∈ V → 𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
473, 46syl 17 . . 3 (𝐺𝐵𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
4847adantl 482 . 2 ((𝐹𝐵𝐺𝐵) → 𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
49 id 22 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
50 oveq2 6623 . . . . . . 7 (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
5149, 50oveqan12d 6634 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 (𝑞 · 𝑔)) = (𝐹 (𝑞 · 𝐺)))
5251fveq2d 6162 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷‘(𝑓 (𝑞 · 𝑔))) = (𝐷‘(𝐹 (𝑞 · 𝐺))))
53 fveq2 6158 . . . . . 6 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
5453adantl 482 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷𝑔) = (𝐷𝐺))
5552, 54breq12d 4636 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔) ↔ (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5655riotabidv 6578 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5756adantl 482 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
58 simpl 473 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
59 simpr 477 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
60 riotaex 6580 . . 3 (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V
6160a1i 11 . 2 ((𝐹𝐵𝐺𝐵) → (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V)
6248, 57, 58, 59, 61ovmpt2d 6753 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝑄𝐺) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3190  ⦋csb 3519   class class class wbr 4623  ‘cfv 5857  ℩crio 6575  (class class class)co 6615   ↦ cmpt2 6617   < clt 10034  Basecbs 15800  .rcmulr 15882  -gcsg 17364  Poly1cpl1 19487   deg1 cdg1 23752  quot1pcq1p 23825 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-slot 15804  df-base 15805  df-q1p 23830 This theorem is referenced by:  q1peqb  23852
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