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Theorem uniioombllem2a 25028
Description: Lemma for uniioombl 25035. (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 481 . . . . . . . . 9 ((𝜑𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
32ffvelcdmda 7071 . . . . . . . 8 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
43elin2d 4195 . . . . . . 7 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ (ℝ × ℝ))
5 1st2nd2 7996 . . . . . . 7 ((𝐹𝑧) ∈ (ℝ × ℝ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
64, 5syl 17 . . . . . 6 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
76fveq2d 6882 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩))
8 df-ov 7396 . . . . 5 ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
97, 8eqtr4di 2789 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))))
10 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1110ffvelcdmda 7071 . . . . . . . . 9 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
1211elin2d 4195 . . . . . . . 8 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ (ℝ × ℝ))
13 1st2nd2 7996 . . . . . . . 8 ((𝐺𝐽) ∈ (ℝ × ℝ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1412, 13syl 17 . . . . . . 7 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1514fveq2d 6882 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩))
16 df-ov 7396 . . . . . 6 ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1715, 16eqtr4di 2789 . . . . 5 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
1817adantr 481 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
199, 18ineq12d 4209 . . 3 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) = (((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) ∩ ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽)))))
20 ovolfcl 24912 . . . . . . 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 1142 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ)
2322rexrd 11246 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ*)
2421simp2d 1143 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ)
2524rexrd 11246 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ*)
26 ovolfcl 24912 . . . . . . . 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 1142 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ)
2928rexrd 11246 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3029adantr 481 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3127simp2d 1143 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ)
3231rexrd 11246 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
3332adantr 481 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
34 iooin 13340 . . . 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 2771 . 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 13410 . 2 (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))) ∈ ran (,)
3836, 37eqeltrdi 2840 1 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) ∈ ran (,))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cin 3943  wss 3944  ifcif 4522  cop 4628   cuni 4901  Disj wdisj 5106   class class class wbr 5141   × cxp 5667  ran crn 5670  ccom 5673  wf 6528  cfv 6532  (class class class)co 7393  1st c1st 7955  2nd c2nd 7956  supcsup 9417  cr 11091  1c1 11093   + caddc 11095  *cxr 11229   < clt 11230  cle 11231  cmin 11426  cn 12194  +crp 12956  (,)cioo 13306  seqcseq 13948  abscabs 15163  vol*covol 24908
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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-sup 9419  df-inf 9420  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-n0 12455  df-z 12541  df-uz 12805  df-q 12915  df-ioo 13310
This theorem is referenced by:  uniioombllem2  25029
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