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Theorem ovolfcl 24965
Description: Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolfcl ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))

Proof of Theorem ovolfcl
StepHypRef Expression
1 ffvelcdm 7079 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
21elin2d 4198 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
3 1st2nd2 8009 . . . 4 ((𝐹𝑁) ∈ (ℝ × ℝ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
42, 3syl 17 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
54, 1eqeltrrd 2835 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
6 ancom 462 . . 3 (((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)) ↔ (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ) ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
7 elin 3963 . . . 4 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)))
8 df-br 5148 . . . . . 6 ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ↔ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ )
98bicomi 223 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ↔ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)))
10 opelxp 5711 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ) ↔ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ))
119, 10anbi12i 628 . . . 4 ((⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)))
127, 11bitri 275 . . 3 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)))
13 df-3an 1090 . . 3 (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))) ↔ (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ) ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
146, 12, 133bitr4i 303 . 2 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
155, 14sylib 217 1 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cin 3946  cop 4633   class class class wbr 5147   × cxp 5673  wf 6536  cfv 6540  1st c1st 7968  2nd c2nd 7969  cr 11105  cle 11245  cn 12208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-1st 7970  df-2nd 7971
This theorem is referenced by:  ovolfioo  24966  ovolficc  24967  ovolfsval  24969  ovolfsf  24970  ovollb2lem  24987  ovolshftlem1  25008  ovolscalem1  25012  ioombl1lem1  25057  ioombl1lem3  25059  ioombl1lem4  25060  ovolfs2  25070  uniiccdif  25077  uniioovol  25078  uniioombllem2a  25081  uniioombllem2  25082  uniioombllem3a  25083  uniioombllem3  25084  uniioombllem4  25085  uniioombllem6  25087  ovolval3  45298
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