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Theorem ovolicc2lem1 23092
 Description: Lemma for ovolicc2 23097. (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 6267 . . . . 5 ((𝜑𝑋𝑈) → (𝐺𝑋) ∈ ℕ)
3 ovolicc2.5 . . . . . . 7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4 inss2 3796 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
5 fss 5969 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
63, 4, 5sylancl 693 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
7 fvco3 6185 . . . . . 6 ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
86, 7sylan 487 . . . . 5 ((𝜑 ∧ (𝐺𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
92, 8syldan 486 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
10 ovolicc2.9 . . . . . 6 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
1110ralrimiva 2949 . . . . 5 (𝜑 → ∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
12 fveq2 6103 . . . . . . . 8 (𝑡 = 𝑋 → (𝐺𝑡) = (𝐺𝑋))
1312fveq2d 6107 . . . . . . 7 (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺𝑡)) = (((,) ∘ 𝐹)‘(𝐺𝑋)))
14 id 22 . . . . . . 7 (𝑡 = 𝑋𝑡 = 𝑋)
1513, 14eqeq12d 2625 . . . . . 6 (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋))
1615rspccva 3281 . . . . 5 ((∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
1711, 16sylan 487 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
186adantr 480 . . . . . . . 8 ((𝜑𝑋𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
1918, 2ffvelrnd 6268 . . . . . . 7 ((𝜑𝑋𝑈) → (𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ))
20 1st2nd2 7096 . . . . . . 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 6107 . . . . 5 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩))
23 df-ov 6552 . . . . 5 ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
2422, 23syl6eqr 2662 . . . 4 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
259, 17, 243eqtr3d 2652 . . 3 ((𝜑𝑋𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
2625eleq2d 2673 . 2 ((𝜑𝑋𝑈) → (𝑃𝑋𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋))))))
27 xp1st 7089 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2819, 27syl 17 . . 3 ((𝜑𝑋𝑈) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
29 xp2nd 7090 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
3019, 29syl 17 . . 3 ((𝜑𝑋𝑈) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
31 rexr 9964 . . . 4 ((1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
32 rexr 9964 . . . 4 ((2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
33 elioo2 12087 . . . 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 493 . . 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 691 . 2 ((𝜑𝑋𝑈) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
3626, 35bitrd 267 1 ((𝜑𝑋𝑈) → (𝑃𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  ℝcr 9814  1c1 9816   + caddc 9818  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  (,)cioo 12046  [,]cicc 12049  seqcseq 12663  abscabs 13822 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-ioo 12050 This theorem is referenced by:  ovolicc2lem2  23093  ovolicc2lem3  23094  ovolicc2lem4  23095
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