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Theorem q1pval 24746
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 16543 . . . 4 (𝐺𝐵𝑅 ∈ V)
4 q1pval.q . . . . 5 𝑄 = (quot1p𝑅)
5 fveq2 6669 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65, 1syl6eqr 2874 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
76csbeq1d 3886 . . . . . . 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))))
81fvexi 6683 . . . . . . . . 9 𝑃 ∈ V
98a1i 11 . . . . . . . 8 (𝑟 = 𝑅𝑃 ∈ V)
10 fveq2 6669 . . . . . . . . . . . 12 (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
1110adantl 484 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
1211, 2syl6eqr 2874 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
1312csbeq1d 3886 . . . . . . . . 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))))
142fvexi 6683 . . . . . . . . . . 11 𝐵 ∈ V
1514a1i 11 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → 𝐵 ∈ V)
16 simpr 487 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
17 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
1817ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = ( deg1𝑅))
19 q1pval.d . . . . . . . . . . . . . . 15 𝐷 = ( deg1𝑅)
2018, 19syl6eqr 2874 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = 𝐷)
21 fveq2 6669 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑃 → (-g𝑝) = (-g𝑃))
2221ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = (-g𝑃))
23 q1pval.m . . . . . . . . . . . . . . . 16 = (-g𝑃)
2422, 23syl6eqr 2874 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = )
25 eqidd 2822 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
26 fveq2 6669 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑃 → (.r𝑝) = (.r𝑃))
2726ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = (.r𝑃))
28 q1pval.t . . . . . . . . . . . . . . . . 17 · = (.r𝑃)
2927, 28syl6eqr 2874 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = · )
3029oveqd 7172 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r𝑝)𝑔) = (𝑞 · 𝑔))
3124, 25, 30oveq123d 7176 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g𝑝)(𝑞(.r𝑝)𝑔)) = (𝑓 (𝑞 · 𝑔)))
3220, 31fveq12d 6676 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) = (𝐷‘(𝑓 (𝑞 · 𝑔))))
3320fveq1d 6671 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
3432, 33breq12d 5078 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔) ↔ (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3516, 34riotaeqbidv 7116 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔)) = (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3616, 16, 35mpoeq123dv 7228 . . . . . . . . . 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))))
3715, 36csbied 3918 . . . . . . . . 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))))
3813, 37eqtrd 2856 . . . . . . . 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))))
399, 38csbied 3918 . . . . . . 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))))
407, 39eqtrd 2856 . . . . . 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))))
41 df-q1p 24725 . . . . . 6 quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
4214, 14mpoex 7776 . . . . . 6 (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))) ∈ V
4340, 41, 42fvmpt 6767 . . . . 5 (𝑅 ∈ V → (quot1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
444, 43syl5eq 2868 . . . 4 (𝑅 ∈ V → 𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
453, 44syl 17 . . 3 (𝐺𝐵𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
4645adantl 484 . 2 ((𝐹𝐵𝐺𝐵) → 𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
47 id 22 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
48 oveq2 7163 . . . . . . 7 (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
4947, 48oveqan12d 7174 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 (𝑞 · 𝑔)) = (𝐹 (𝑞 · 𝐺)))
5049fveq2d 6673 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷‘(𝑓 (𝑞 · 𝑔))) = (𝐷‘(𝐹 (𝑞 · 𝐺))))
51 fveq2 6669 . . . . . 6 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
5251adantl 484 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷𝑔) = (𝐷𝐺))
5350, 52breq12d 5078 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔) ↔ (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5453riotabidv 7115 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5554adantl 484 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
56 simpl 485 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
57 simpr 487 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
58 riotaex 7117 . . 3 (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V
5958a1i 11 . 2 ((𝐹𝐵𝐺𝐵) → (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V)
6046, 55, 56, 57, 59ovmpod 7301 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝑄𝐺) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  csb 3882   class class class wbr 5065  cfv 6354  crio 7112  (class class class)co 7155  cmpo 7157   < clt 10674  Basecbs 16482  .rcmulr 16565  -gcsg 18104  Poly1cpl1 20344   deg1 cdg1 24647  quot1pcq1p 24720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-slot 16486  df-base 16488  df-q1p 24725
This theorem is referenced by:  q1peqb  24747
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