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Theorem ovolicc2lem1 23504
Description: Lemma for ovolicc2 23509. (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 ovolicc2.8 . . . . . 6 (𝜑𝐺:𝑈⟶ℕ)
21ffvelrnda 6502 . . . . 5 ((𝜑𝑋𝑈) → (𝐺𝑋) ∈ ℕ)
3 ovolicc2.5 . . . . . . 7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4 inss2 3980 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
5 fss 6196 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
63, 4, 5sylancl 566 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
7 fvco3 6417 . . . . . 6 ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
86, 7sylan 561 . . . . 5 ((𝜑 ∧ (𝐺𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
92, 8syldan 571 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
10 ovolicc2.9 . . . . . 6 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
1110ralrimiva 3114 . . . . 5 (𝜑 → ∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
12 fveq2 6332 . . . . . . . 8 (𝑡 = 𝑋 → (𝐺𝑡) = (𝐺𝑋))
1312fveq2d 6336 . . . . . . 7 (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺𝑡)) = (((,) ∘ 𝐹)‘(𝐺𝑋)))
14 id 22 . . . . . . 7 (𝑡 = 𝑋𝑡 = 𝑋)
1513, 14eqeq12d 2785 . . . . . 6 (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋))
1615rspccva 3457 . . . . 5 ((∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
1711, 16sylan 561 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
186adantr 466 . . . . . . . 8 ((𝜑𝑋𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
1918, 2ffvelrnd 6503 . . . . . . 7 ((𝜑𝑋𝑈) → (𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ))
20 1st2nd2 7353 . . . . . . 7 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺𝑋)) = ⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
2119, 20syl 17 . . . . . 6 ((𝜑𝑋𝑈) → (𝐹‘(𝐺𝑋)) = ⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
2221fveq2d 6336 . . . . 5 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩))
23 df-ov 6795 . . . . 5 ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
2422, 23syl6eqr 2822 . . . 4 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
259, 17, 243eqtr3d 2812 . . 3 ((𝜑𝑋𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
2625eleq2d 2835 . 2 ((𝜑𝑋𝑈) → (𝑃𝑋𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋))))))
27 xp1st 7346 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2819, 27syl 17 . . 3 ((𝜑𝑋𝑈) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
29 xp2nd 7347 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
3019, 29syl 17 . . 3 ((𝜑𝑋𝑈) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
31 rexr 10286 . . . 4 ((1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
32 rexr 10286 . . . 4 ((2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
33 elioo2 12420 . . . 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))))))
3431, 32, 33syl2an 575 . . 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))))))
3528, 30, 34syl2anc 565 . 2 ((𝜑𝑋𝑈) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
3626, 35bitrd 268 1 ((𝜑𝑋𝑈) → (𝑃𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  cin 3720  wss 3721  𝒫 cpw 4295  cop 4320   cuni 4572   class class class wbr 4784   × cxp 5247  ran crn 5250  ccom 5253  wf 6027  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  Fincfn 8108  cr 10136  1c1 10138   + caddc 10140  *cxr 10274   < clt 10275  cle 10276  cmin 10467  cn 11221  (,)cioo 12379  [,]cicc 12382  seqcseq 13007  abscabs 14181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-pre-lttri 10211  ax-pre-lttrn 10212
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-ioo 12383
This theorem is referenced by:  ovolicc2lem2  23505  ovolicc2lem3  23506  ovolicc2lem4  23507
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