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Theorem ovolicc2lem1 24881
Description: Lemma for ovolicc2 24886. (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 4189 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 fss 6685 . . . . . 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 7035 . . . . 5 ((𝜑𝑋𝑈) → (𝐺𝑋) ∈ ℕ)
7 fvco3 6940 . . . . 5 ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
84, 6, 7syl2an2r 683 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
9 ovolicc2.9 . . . . . 6 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
109ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
11 2fveq3 6847 . . . . . . 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 3580 . . . . 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 7036 . . . . . . 7 ((𝜑𝑋𝑈) → (𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ))
18 1st2nd2 7960 . . . . . . 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 6846 . . . . 5 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩))
21 df-ov 7360 . . . . 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 2823 . 2 ((𝜑𝑋𝑈) → (𝑃𝑋𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋))))))
25 xp1st 7953 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2617, 25syl 17 . . 3 ((𝜑𝑋𝑈) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
27 xp2nd 7954 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2817, 27syl 17 . . 3 ((𝜑𝑋𝑈) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
29 rexr 11201 . . . 4 ((1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
30 rexr 11201 . . . 4 ((2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
31 elioo2 13305 . . . 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 1087   = wceq 1541  wcel 2106  wral 3064  cin 3909  wss 3910  𝒫 cpw 4560  cop 4592   cuni 4865   class class class wbr 5105   × cxp 5631  ran crn 5634  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Fincfn 8883  cr 11050  1c1 11052   + caddc 11054  *cxr 11188   < clt 11189  cle 11190  cmin 11385  cn 12153  (,)cioo 13264  [,]cicc 13267  seqcseq 13906  abscabs 15119
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-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-pre-lttri 11125  ax-pre-lttrn 11126
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-nel 3050  df-ral 3065  df-rex 3074  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-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-id 5531  df-po 5545  df-so 5546  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-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-ioo 13268
This theorem is referenced by:  ovolicc2lem2  24882  ovolicc2lem3  24883  ovolicc2lem4  24884
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