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Theorem q1pval 24154
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 16147 . . . 4 (𝐺𝐵𝑅 ∈ V)
4 q1pval.q . . . . 5 𝑄 = (quot1p𝑅)
5 fveq2 6348 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65, 1syl6eqr 2826 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
76csbeq1d 3695 . . . . . . 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 6360 . . . . . . . . 9 𝑃 ∈ V
98a1i 11 . . . . . . . 8 (𝑟 = 𝑅𝑃 ∈ V)
10 fveq2 6348 . . . . . . . . . . . 12 (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
1110adantl 468 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
1211, 2syl6eqr 2826 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
1312csbeq1d 3695 . . . . . . . . 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 6360 . . . . . . . . . . 11 𝐵 ∈ V
1514a1i 11 . . . . . . . . . 10 ((𝑟 = 𝑅𝑝 = 𝑃) → 𝐵 ∈ V)
16 simpr 472 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
17 fveq2 6348 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
1817ad2antrr 706 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = ( deg1𝑅))
19 q1pval.d . . . . . . . . . . . . . . 15 𝐷 = ( deg1𝑅)
2018, 19syl6eqr 2826 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1𝑟) = 𝐷)
21 fveq2 6348 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑃 → (-g𝑝) = (-g𝑃))
2221ad2antlr 707 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = (-g𝑃))
23 q1pval.m . . . . . . . . . . . . . . . 16 = (-g𝑃)
2422, 23syl6eqr 2826 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g𝑝) = )
25 eqidd 2775 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
26 fveq2 6348 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑃 → (.r𝑝) = (.r𝑃))
2726ad2antlr 707 . . . . . . . . . . . . . . . . 17 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = (.r𝑃))
28 q1pval.t . . . . . . . . . . . . . . . . 17 · = (.r𝑃)
2927, 28syl6eqr 2826 . . . . . . . . . . . . . . . 16 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r𝑝) = · )
3029oveqd 6829 . . . . . . . . . . . . . . 15 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r𝑝)𝑔) = (𝑞 · 𝑔))
3124, 25, 30oveq123d 6833 . . . . . . . . . . . . . 14 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g𝑝)(𝑞(.r𝑝)𝑔)) = (𝑓 (𝑞 · 𝑔)))
3220, 31fveq12d 6355 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) = (𝐷‘(𝑓 (𝑞 · 𝑔))))
3320fveq1d 6350 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
3432, 33breq12d 4810 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔) ↔ (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3516, 34riotaeqbidv 6776 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔)) = (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)))
3616, 16, 35mpt2eq123dv 6885 . . . . . . . . . 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 3715 . . . . . . . . 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 2808 . . . . . . . 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 3715 . . . . . . 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 2808 . . . . . 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 24133 . . . . . 6 quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
4214, 14mpt2ex 7418 . . . . . 6 (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))) ∈ V
4340, 41, 42fvmpt 6441 . . . . 5 (𝑅 ∈ V → (quot1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
444, 43syl5eq 2820 . . . 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 468 . 2 ((𝐹𝐵𝐺𝐵) → 𝑄 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔))))
47 id 22 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
48 oveq2 6820 . . . . . . 7 (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
4947, 48oveqan12d 6831 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 (𝑞 · 𝑔)) = (𝐹 (𝑞 · 𝐺)))
5049fveq2d 6352 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷‘(𝑓 (𝑞 · 𝑔))) = (𝐷‘(𝐹 (𝑞 · 𝐺))))
51 fveq2 6348 . . . . . 6 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
5251adantl 468 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐷𝑔) = (𝐷𝐺))
5350, 52breq12d 4810 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔) ↔ (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5453riotabidv 6775 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
5554adantl 468 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑞𝐵 (𝐷‘(𝑓 (𝑞 · 𝑔))) < (𝐷𝑔)) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
56 simpl 469 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
57 simpr 472 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
58 riotaex 6777 . . 3 (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V
5958a1i 11 . 2 ((𝐹𝐵𝐺𝐵) → (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)) ∈ V)
6046, 55, 56, 57, 59ovmpt2d 6956 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝑄𝐺) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1634  wcel 2148  Vcvv 3355  csb 3688   class class class wbr 4797  cfv 6042  crio 6772  (class class class)co 6812  cmpt2 6814   < clt 10297  Basecbs 16084  .rcmulr 16170  -gcsg 17652  Poly1cpl1 19782   deg1 cdg1 24055  quot1pcq1p 24128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-1st 7336  df-2nd 7337  df-slot 16088  df-base 16090  df-q1p 24133
This theorem is referenced by:  q1peqb  24155
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