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Theorem q1pval 23958
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 15967 . . . 4 (𝐺𝐵𝑅 ∈ V)
4 q1pval.q . . . . 5 𝑄 = (quot1p𝑅)
5 fveq2 6229 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65, 1syl6eqr 2703 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
76csbeq1d 3573 . . . . . . 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 6239 . . . . . . . . . 10 (Poly1𝑅) ∈ V
91, 8eqeltri 2726 . . . . . . . . 9 𝑃 ∈ V
109a1i 11 . . . . . . . 8 (𝑟 = 𝑅𝑃 ∈ V)
11 fveq2 6229 . . . . . . . . . . . 12 (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
1211adantl 481 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
1312, 2syl6eqr 2703 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
1413csbeq1d 3573 . . . . . . . . 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 6239 . . . . . . . . . . . 12 (Base‘𝑃) ∈ V
162, 15eqeltri 2726 . . . . . . . . . . 11 𝐵 ∈ V
1716a1i 11 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → 𝐵 ∈ V)
18 simpr 476 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
19 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
2019ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = ( deg1𝑅))
21 q1pval.d . . . . . . . . . . . . . . 15 𝐷 = ( deg1𝑅)
2220, 21syl6eqr 2703 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = 𝐷)
23 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑃 → (-g𝑝) = (-g𝑃))
2423ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = (-g𝑃))
25 q1pval.m . . . . . . . . . . . . . . . 16 = (-g𝑃)
2624, 25syl6eqr 2703 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = )
27 eqidd 2652 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
28 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑃 → (.r𝑝) = (.r𝑃))
2928ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = (.r𝑃))
30 q1pval.t . . . . . . . . . . . . . . . . 17 · = (.r𝑃)
3129, 30syl6eqr 2703 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = · )
3231oveqd 6707 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r𝑝)𝑔) = (𝑞 · 𝑔))
3326, 27, 32oveq123d 6711 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g𝑝)(𝑞(.r𝑝)𝑔)) = (𝑓 (𝑞 · 𝑔)))
3422, 33fveq12d 6235 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) = (𝐷‘(𝑓 (𝑞 · 𝑔))))
3522fveq1d 6231 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
3634, 35breq12d 4698 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔) ↔ (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3718, 36riotaeqbidv 6654 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔)) = (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3818, 18, 37mpt2eq123dv 6759 . . . . . . . . . 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 3593 . . . . . . . . 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 2685 . . . . . . . 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 3593 . . . . . . 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 2685 . . . . . 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 23937 . . . . . 6 quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
4416, 16mpt2ex 7292 . . . . . 6 (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))) ∈ V
4542, 43, 44fvmpt 6321 . . . . 5 (𝑅 ∈ V → (quot1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
464, 45syl5eq 2697 . . . 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 481 . 2 ((𝐹𝐵𝐺𝐵) → 𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
49 id 22 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
50 oveq2 6698 . . . . . . 7 (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
5149, 50oveqan12d 6709 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 (𝑞 · 𝑔)) = (𝐹 (𝑞 · 𝐺)))
5251fveq2d 6233 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷‘(𝑓 (𝑞 · 𝑔))) = (𝐷‘(𝐹 (𝑞 · 𝐺))))
53 fveq2 6229 . . . . . 6 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
5453adantl 481 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷𝑔) = (𝐷𝐺))
5552, 54breq12d 4698 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔) ↔ (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5655riotabidv 6653 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5756adantl 481 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
58 simpl 472 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
59 simpr 476 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
60 riotaex 6655 . . 3 (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V
6160a1i 11 . 2 ((𝐹𝐵𝐺𝐵) → (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V)
6248, 57, 58, 59, 61ovmpt2d 6830 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝑄𝐺) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  csb 3566   class class class wbr 4685  cfv 5926  crio 6650  (class class class)co 6690  cmpt2 6692   < clt 10112  Basecbs 15904  .rcmulr 15989  -gcsg 17471  Poly1cpl1 19595   deg1 cdg1 23859  quot1pcq1p 23932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-slot 15908  df-base 15910  df-q1p 23937
This theorem is referenced by:  q1peqb  23959
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