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Theorem q1pval 25518
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 17089 . . . 4 (𝐺𝐵𝑅 ∈ V)
4 q1pval.q . . . . 5 𝑄 = (quot1p𝑅)
5 fveq2 6842 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65, 1eqtr4di 2794 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
76csbeq1d 3859 . . . . . . 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 6856 . . . . . . . . 9 𝑃 ∈ V
98a1i 11 . . . . . . . 8 (𝑟 = 𝑅𝑃 ∈ V)
10 fveq2 6842 . . . . . . . . . . . 12 (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
1110adantl 482 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
1211, 2eqtr4di 2794 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
1312csbeq1d 3859 . . . . . . . . 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 6856 . . . . . . . . . . 11 𝐵 ∈ V
1514a1i 11 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → 𝐵 ∈ V)
16 simpr 485 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
17 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
1817ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = ( deg1𝑅))
19 q1pval.d . . . . . . . . . . . . . . 15 𝐷 = ( deg1𝑅)
2018, 19eqtr4di 2794 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = 𝐷)
21 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑃 → (-g𝑝) = (-g𝑃))
2221ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = (-g𝑃))
23 q1pval.m . . . . . . . . . . . . . . . 16 = (-g𝑃)
2422, 23eqtr4di 2794 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = )
25 eqidd 2737 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
26 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑃 → (.r𝑝) = (.r𝑃))
2726ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = (.r𝑃))
28 q1pval.t . . . . . . . . . . . . . . . . 17 · = (.r𝑃)
2927, 28eqtr4di 2794 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = · )
3029oveqd 7374 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r𝑝)𝑔) = (𝑞 · 𝑔))
3124, 25, 30oveq123d 7378 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g𝑝)(𝑞(.r𝑝)𝑔)) = (𝑓 (𝑞 · 𝑔)))
3220, 31fveq12d 6849 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) = (𝐷‘(𝑓 (𝑞 · 𝑔))))
3320fveq1d 6844 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
3432, 33breq12d 5118 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔) ↔ (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3516, 34riotaeqbidv 7316 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔)) = (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3616, 16, 35mpoeq123dv 7432 . . . . . . . . . 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 3893 . . . . . . . . 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 2776 . . . . . . . 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 3893 . . . . . . 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 2776 . . . . . 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 25497 . . . . . 6 quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
4214, 14mpoex 8012 . . . . . 6 (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))) ∈ V
4340, 41, 42fvmpt 6948 . . . . 5 (𝑅 ∈ V → (quot1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
444, 43eqtrid 2788 . . . 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 482 . 2 ((𝐹𝐵𝐺𝐵) → 𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
47 id 22 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
48 oveq2 7365 . . . . . . 7 (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
4947, 48oveqan12d 7376 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 (𝑞 · 𝑔)) = (𝐹 (𝑞 · 𝐺)))
5049fveq2d 6846 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷‘(𝑓 (𝑞 · 𝑔))) = (𝐷‘(𝐹 (𝑞 · 𝐺))))
51 fveq2 6842 . . . . . 6 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
5251adantl 482 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷𝑔) = (𝐷𝐺))
5350, 52breq12d 5118 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔) ↔ (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5453riotabidv 7315 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5554adantl 482 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
56 simpl 483 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
57 simpr 485 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
58 riotaex 7317 . . 3 (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V
5958a1i 11 . 2 ((𝐹𝐵𝐺𝐵) → (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V)
6046, 55, 56, 57, 59ovmpod 7507 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝑄𝐺) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  csb 3855   class class class wbr 5105  cfv 6496  crio 7312  (class class class)co 7357  cmpo 7359   < clt 11189  Basecbs 17083  .rcmulr 17134  -gcsg 18750  Poly1cpl1 21548   deg1 cdg1 25416  quot1pcq1p 25492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-slot 17054  df-ndx 17066  df-base 17084  df-q1p 25497
This theorem is referenced by:  q1peqb  25519
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