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Theorem ovolicc2lem1 24787
Description: Lemma for ovolicc2 24792. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1 (𝜑𝐴 ∈ ℝ)
ovolicc.2 (𝜑𝐵 ∈ ℝ)
ovolicc.3 (𝜑𝐴𝐵)
ovolicc2.4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolicc2.5 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolicc2.6 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
ovolicc2.8 (𝜑𝐺:𝑈⟶ℕ)
ovolicc2.9 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
Assertion
Ref Expression
ovolicc2lem1 ((𝜑𝑋𝑈) → (𝑃𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
Distinct variable groups:   𝑡,𝐴   𝑡,𝐵   𝑡,𝐹   𝑡,𝐺   𝜑,𝑡   𝑡,𝑈   𝑡,𝑋
Allowed substitution hints:   𝑃(𝑡)   𝑆(𝑡)

Proof of Theorem ovolicc2lem1
StepHypRef Expression
1 ovolicc2.5 . . . . . 6 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2 inss2 4176 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 fss 6668 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
41, 2, 3sylancl 586 . . . . 5 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
5 ovolicc2.8 . . . . . 6 (𝜑𝐺:𝑈⟶ℕ)
65ffvelcdmda 7017 . . . . 5 ((𝜑𝑋𝑈) → (𝐺𝑋) ∈ ℕ)
7 fvco3 6923 . . . . 5 ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
84, 6, 7syl2an2r 682 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
9 ovolicc2.9 . . . . . 6 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
109ralrimiva 3139 . . . . 5 (𝜑 → ∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
11 2fveq3 6830 . . . . . . 7 (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺𝑡)) = (((,) ∘ 𝐹)‘(𝐺𝑋)))
12 id 22 . . . . . . 7 (𝑡 = 𝑋𝑡 = 𝑋)
1311, 12eqeq12d 2752 . . . . . 6 (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋))
1413rspccva 3569 . . . . 5 ((∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
1510, 14sylan 580 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
164adantr 481 . . . . . . . 8 ((𝜑𝑋𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
1716, 6ffvelcdmd 7018 . . . . . . 7 ((𝜑𝑋𝑈) → (𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ))
18 1st2nd2 7938 . . . . . . 7 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺𝑋)) = ⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
1917, 18syl 17 . . . . . 6 ((𝜑𝑋𝑈) → (𝐹‘(𝐺𝑋)) = ⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
2019fveq2d 6829 . . . . 5 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩))
21 df-ov 7340 . . . . 5 ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
2220, 21eqtr4di 2794 . . . 4 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
238, 15, 223eqtr3d 2784 . . 3 ((𝜑𝑋𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
2423eleq2d 2822 . 2 ((𝜑𝑋𝑈) → (𝑃𝑋𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋))))))
25 xp1st 7931 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2617, 25syl 17 . . 3 ((𝜑𝑋𝑈) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
27 xp2nd 7932 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2817, 27syl 17 . . 3 ((𝜑𝑋𝑈) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
29 rexr 11122 . . . 4 ((1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
30 rexr 11122 . . . 4 ((2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
31 elioo2 13221 . . . 4 (((1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ* ∧ (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
3229, 30, 31syl2an 596 . . 3 (((1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
3326, 28, 32syl2anc 584 . 2 ((𝜑𝑋𝑈) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
3424, 33bitrd 278 1 ((𝜑𝑋𝑈) → (𝑃𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3061  cin 3897  wss 3898  𝒫 cpw 4547  cop 4579   cuni 4852   class class class wbr 5092   × cxp 5618  ran crn 5621  ccom 5624  wf 6475  cfv 6479  (class class class)co 7337  1st c1st 7897  2nd c2nd 7898  Fincfn 8804  cr 10971  1c1 10973   + caddc 10975  *cxr 11109   < clt 11110  cle 11111  cmin 11306  cn 12074  (,)cioo 13180  [,]cicc 13183  seqcseq 13822  abscabs 15044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-pre-lttri 11046  ax-pre-lttrn 11047
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-ioo 13184
This theorem is referenced by:  ovolicc2lem2  24788  ovolicc2lem3  24789  ovolicc2lem4  24790
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