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Theorem uniioombllem2a 24227
 Description: Lemma for uniioombl 24234. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
uniioombl.1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
uniioombl.3 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
uniioombl.a 𝐴 = ran ((,) ∘ 𝐹)
uniioombl.e (𝜑 → (vol*‘𝐸) ∈ ℝ)
uniioombl.c (𝜑𝐶 ∈ ℝ+)
uniioombl.g (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.s (𝜑𝐸 ran ((,) ∘ 𝐺))
uniioombl.t 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
uniioombl.v (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
Assertion
Ref Expression
uniioombllem2a (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) ∈ ran (,))
Distinct variable groups:   𝑥,𝑧,𝐹   𝑥,𝐺,𝑧   𝑥,𝐴,𝑧   𝑥,𝐶,𝑧   𝑥,𝐽,𝑧   𝜑,𝑥,𝑧   𝑥,𝑇,𝑧
Allowed substitution hints:   𝑆(𝑥,𝑧)   𝐸(𝑥,𝑧)

Proof of Theorem uniioombllem2a
StepHypRef Expression
1 uniioombl.1 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21adantr 484 . . . . . . . . 9 ((𝜑𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
32ffvelrnda 6838 . . . . . . . 8 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
43elin2d 4129 . . . . . . 7 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ (ℝ × ℝ))
5 1st2nd2 7723 . . . . . . 7 ((𝐹𝑧) ∈ (ℝ × ℝ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
64, 5syl 17 . . . . . 6 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
76fveq2d 6659 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩))
8 df-ov 7148 . . . . 5 ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
97, 8eqtr4di 2851 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))))
10 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1110ffvelrnda 6838 . . . . . . . . 9 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
1211elin2d 4129 . . . . . . . 8 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ (ℝ × ℝ))
13 1st2nd2 7723 . . . . . . . 8 ((𝐺𝐽) ∈ (ℝ × ℝ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1412, 13syl 17 . . . . . . 7 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1514fveq2d 6659 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩))
16 df-ov 7148 . . . . . 6 ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1715, 16eqtr4di 2851 . . . . 5 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
1817adantr 484 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
199, 18ineq12d 4143 . . 3 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) = (((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) ∩ ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽)))))
20 ovolfcl 24111 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑧 ∈ ℕ) → ((1st ‘(𝐹𝑧)) ∈ ℝ ∧ (2nd ‘(𝐹𝑧)) ∈ ℝ ∧ (1st ‘(𝐹𝑧)) ≤ (2nd ‘(𝐹𝑧))))
212, 20sylan 583 . . . . . 6 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((1st ‘(𝐹𝑧)) ∈ ℝ ∧ (2nd ‘(𝐹𝑧)) ∈ ℝ ∧ (1st ‘(𝐹𝑧)) ≤ (2nd ‘(𝐹𝑧))))
2221simp1d 1139 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ)
2322rexrd 10698 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ*)
2421simp2d 1140 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ)
2524rexrd 10698 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ*)
26 ovolfcl 24111 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐽 ∈ ℕ) → ((1st ‘(𝐺𝐽)) ∈ ℝ ∧ (2nd ‘(𝐺𝐽)) ∈ ℝ ∧ (1st ‘(𝐺𝐽)) ≤ (2nd ‘(𝐺𝐽))))
2710, 26sylan 583 . . . . . . 7 ((𝜑𝐽 ∈ ℕ) → ((1st ‘(𝐺𝐽)) ∈ ℝ ∧ (2nd ‘(𝐺𝐽)) ∈ ℝ ∧ (1st ‘(𝐺𝐽)) ≤ (2nd ‘(𝐺𝐽))))
2827simp1d 1139 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ)
2928rexrd 10698 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3029adantr 484 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3127simp2d 1140 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ)
3231rexrd 10698 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
3332adantr 484 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
34 iooin 12780 . . . 4 ((((1st ‘(𝐹𝑧)) ∈ ℝ* ∧ (2nd ‘(𝐹𝑧)) ∈ ℝ*) ∧ ((1st ‘(𝐺𝐽)) ∈ ℝ* ∧ (2nd ‘(𝐺𝐽)) ∈ ℝ*)) → (((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) ∩ ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽)))) = (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))))
3523, 25, 30, 33, 34syl22anc 837 . . 3 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) ∩ ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽)))) = (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))))
3619, 35eqtrd 2833 . 2 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) = (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))))
37 ioorebas 12849 . 2 (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))) ∈ ran (,)
3836, 37eqeltrdi 2898 1 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) ∈ ran (,))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3882   ⊆ wss 3883  ifcif 4428  ⟨cop 4534  ∪ cuni 4804  Disj wdisj 4999   class class class wbr 5034   × cxp 5521  ran crn 5524   ∘ ccom 5527  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  1st c1st 7682  2nd c2nd 7683  supcsup 8906  ℝcr 10543  1c1 10545   + caddc 10547  ℝ*cxr 10681   < clt 10682   ≤ cle 10683   − cmin 10877  ℕcn 11643  ℝ+crp 12397  (,)cioo 12746  seqcseq 13384  abscabs 14605  vol*covol 24107 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  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-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-sup 8908  df-inf 8909  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-n0 11904  df-z 11990  df-uz 12252  df-q 12357  df-ioo 12750 This theorem is referenced by:  uniioombllem2  24228
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