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Theorem uniioombllem2a 25631
Description: Lemma for uniioombl 25638. (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 480 . . . . . . . . 9 ((𝜑𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
32ffvelcdmda 7104 . . . . . . . 8 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
43elin2d 4215 . . . . . . 7 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ (ℝ × ℝ))
5 1st2nd2 8052 . . . . . . 7 ((𝐹𝑧) ∈ (ℝ × ℝ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
64, 5syl 17 . . . . . 6 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
76fveq2d 6911 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩))
8 df-ov 7434 . . . . 5 ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
97, 8eqtr4di 2793 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))))
10 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1110ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
1211elin2d 4215 . . . . . . . 8 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ (ℝ × ℝ))
13 1st2nd2 8052 . . . . . . . 8 ((𝐺𝐽) ∈ (ℝ × ℝ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1412, 13syl 17 . . . . . . 7 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1514fveq2d 6911 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩))
16 df-ov 7434 . . . . . 6 ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1715, 16eqtr4di 2793 . . . . 5 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
1817adantr 480 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
199, 18ineq12d 4229 . . 3 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) = (((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) ∩ ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽)))))
20 ovolfcl 25515 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑧 ∈ ℕ) → ((1st ‘(𝐹𝑧)) ∈ ℝ ∧ (2nd ‘(𝐹𝑧)) ∈ ℝ ∧ (1st ‘(𝐹𝑧)) ≤ (2nd ‘(𝐹𝑧))))
212, 20sylan 580 . . . . . 6 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((1st ‘(𝐹𝑧)) ∈ ℝ ∧ (2nd ‘(𝐹𝑧)) ∈ ℝ ∧ (1st ‘(𝐹𝑧)) ≤ (2nd ‘(𝐹𝑧))))
2221simp1d 1141 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ)
2322rexrd 11309 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ*)
2421simp2d 1142 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ)
2524rexrd 11309 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ*)
26 ovolfcl 25515 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐽 ∈ ℕ) → ((1st ‘(𝐺𝐽)) ∈ ℝ ∧ (2nd ‘(𝐺𝐽)) ∈ ℝ ∧ (1st ‘(𝐺𝐽)) ≤ (2nd ‘(𝐺𝐽))))
2710, 26sylan 580 . . . . . . 7 ((𝜑𝐽 ∈ ℕ) → ((1st ‘(𝐺𝐽)) ∈ ℝ ∧ (2nd ‘(𝐺𝐽)) ∈ ℝ ∧ (1st ‘(𝐺𝐽)) ≤ (2nd ‘(𝐺𝐽))))
2827simp1d 1141 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ)
2928rexrd 11309 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3029adantr 480 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3127simp2d 1142 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ)
3231rexrd 11309 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
3332adantr 480 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
34 iooin 13418 . . . 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 839 . . 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 2775 . 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 13488 . 2 (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))) ∈ ran (,)
3836, 37eqeltrdi 2847 1 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) ∈ ran (,))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963  ifcif 4531  cop 4637   cuni 4912  Disj wdisj 5115   class class class wbr 5148   × cxp 5687  ran crn 5690  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  supcsup 9478  cr 11152  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cle 11294  cmin 11490  cn 12264  +crp 13032  (,)cioo 13384  seqcseq 14039  abscabs 15270  vol*covol 25511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-ioo 13388
This theorem is referenced by:  uniioombllem2  25632
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