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Theorem uniioombllem2a 25098
Description: Lemma for uniioombl 25105. (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 7086 . . . . . . . 8 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
43elin2d 4199 . . . . . . 7 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ (ℝ × ℝ))
5 1st2nd2 8013 . . . . . . 7 ((𝐹𝑧) ∈ (ℝ × ℝ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
64, 5syl 17 . . . . . 6 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
76fveq2d 6895 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩))
8 df-ov 7411 . . . . 5 ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) = ((,)‘⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
97, 8eqtr4di 2790 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹𝑧)) = ((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))))
10 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1110ffvelcdmda 7086 . . . . . . . . 9 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
1211elin2d 4199 . . . . . . . 8 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ (ℝ × ℝ))
13 1st2nd2 8013 . . . . . . . 8 ((𝐺𝐽) ∈ (ℝ × ℝ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1412, 13syl 17 . . . . . . 7 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1514fveq2d 6895 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩))
16 df-ov 7411 . . . . . 6 ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1715, 16eqtr4di 2790 . . . . 5 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
1817adantr 481 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
199, 18ineq12d 4213 . . 3 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) = (((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) ∩ ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽)))))
20 ovolfcl 24982 . . . . . . 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 11263 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ*)
2421simp2d 1143 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ)
2524rexrd 11263 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ*)
26 ovolfcl 24982 . . . . . . . 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 11263 . . . . 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 11263 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
3332adantr 481 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
34 iooin 13357 . . . 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 2772 . 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 13427 . 2 (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))) ∈ ran (,)
3836, 37eqeltrdi 2841 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 3947  wss 3948  ifcif 4528  cop 4634   cuni 4908  Disj wdisj 5113   class class class wbr 5148   × cxp 5674  ran crn 5677  ccom 5680  wf 6539  cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  supcsup 9434  cr 11108  1c1 11110   + caddc 11112  *cxr 11246   < clt 11247  cle 11248  cmin 11443  cn 12211  +crp 12973  (,)cioo 13323  seqcseq 13965  abscabs 15180  vol*covol 24978
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-q 12932  df-ioo 13327
This theorem is referenced by:  uniioombllem2  25099
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