MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolicc2lem1 Structured version   Visualization version   GIF version

Theorem ovolicc2lem1 24124
Description: Lemma for ovolicc2 24129. (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 4159 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 fss 6505 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
41, 2, 3sylancl 589 . . . . 5 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
5 ovolicc2.8 . . . . . 6 (𝜑𝐺:𝑈⟶ℕ)
65ffvelrnda 6832 . . . . 5 ((𝜑𝑋𝑈) → (𝐺𝑋) ∈ ℕ)
7 fvco3 6741 . . . . 5 ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
84, 6, 7syl2an2r 684 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = ((,)‘(𝐹‘(𝐺𝑋))))
9 ovolicc2.9 . . . . . 6 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
109ralrimiva 3152 . . . . 5 (𝜑 → ∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
11 2fveq3 6654 . . . . . . 7 (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺𝑡)) = (((,) ∘ 𝐹)‘(𝐺𝑋)))
12 id 22 . . . . . . 7 (𝑡 = 𝑋𝑡 = 𝑋)
1311, 12eqeq12d 2817 . . . . . 6 (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋))
1413rspccva 3573 . . . . 5 ((∀𝑡𝑈 (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
1510, 14sylan 583 . . . 4 ((𝜑𝑋𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑋)) = 𝑋)
164adantr 484 . . . . . . . 8 ((𝜑𝑋𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
1716, 6ffvelrnd 6833 . . . . . . 7 ((𝜑𝑋𝑈) → (𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ))
18 1st2nd2 7714 . . . . . . 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 6653 . . . . 5 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩))
21 df-ov 7142 . . . . 5 ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑋))), (2nd ‘(𝐹‘(𝐺𝑋)))⟩)
2220, 21eqtr4di 2854 . . . 4 ((𝜑𝑋𝑈) → ((,)‘(𝐹‘(𝐺𝑋))) = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
238, 15, 223eqtr3d 2844 . . 3 ((𝜑𝑋𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))))
2423eleq2d 2878 . 2 ((𝜑𝑋𝑈) → (𝑃𝑋𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋))))))
25 xp1st 7707 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2617, 25syl 17 . . 3 ((𝜑𝑋𝑈) → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
27 xp2nd 7708 . . . 4 ((𝐹‘(𝐺𝑋)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
2817, 27syl 17 . . 3 ((𝜑𝑋𝑈) → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ)
29 rexr 10680 . . . 4 ((1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
30 rexr 10680 . . . 4 ((2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑋))) ∈ ℝ*)
31 elioo2 12771 . . . 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 598 . . 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 587 . 2 ((𝜑𝑋𝑈) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺𝑋)))(,)(2nd ‘(𝐹‘(𝐺𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
3424, 33bitrd 282 1 ((𝜑𝑋𝑈) → (𝑃𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑋))) < 𝑃𝑃 < (2nd ‘(𝐹‘(𝐺𝑋))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  cin 3883  wss 3884  𝒫 cpw 4500  cop 4534   cuni 4803   class class class wbr 5033   × cxp 5521  ran crn 5524  ccom 5527  wf 6324  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Fincfn 8496  cr 10529  1c1 10531   + caddc 10533  *cxr 10667   < clt 10668  cle 10669  cmin 10863  cn 11629  (,)cioo 12730  [,]cicc 12733  seqcseq 13368  abscabs 14588
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-pre-lttri 10604  ax-pre-lttrn 10605
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-ioo 12734
This theorem is referenced by:  ovolicc2lem2  24125  ovolicc2lem3  24126  ovolicc2lem4  24127
  Copyright terms: Public domain W3C validator