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Theorem uniioombllem2a 25617
Description: Lemma for uniioombl 25624. (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 4205 . . . . . . 7 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) ∈ (ℝ × ℝ))
5 1st2nd2 8053 . . . . . . 7 ((𝐹𝑧) ∈ (ℝ × ℝ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
64, 5syl 17 . . . . . 6 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹𝑧) = ⟨(1st ‘(𝐹𝑧)), (2nd ‘(𝐹𝑧))⟩)
76fveq2d 6910 . . . . 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 2795 . . . 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 4205 . . . . . . . 8 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) ∈ (ℝ × ℝ))
13 1st2nd2 8053 . . . . . . . 8 ((𝐺𝐽) ∈ (ℝ × ℝ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1412, 13syl 17 . . . . . . 7 ((𝜑𝐽 ∈ ℕ) → (𝐺𝐽) = ⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1514fveq2d 6910 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩))
16 df-ov 7434 . . . . . 6 ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))) = ((,)‘⟨(1st ‘(𝐺𝐽)), (2nd ‘(𝐺𝐽))⟩)
1715, 16eqtr4di 2795 . . . . 5 ((𝜑𝐽 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
1817adantr 480 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺𝐽)) = ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽))))
199, 18ineq12d 4221 . . 3 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) = (((1st ‘(𝐹𝑧))(,)(2nd ‘(𝐹𝑧))) ∩ ((1st ‘(𝐺𝐽))(,)(2nd ‘(𝐺𝐽)))))
20 ovolfcl 25501 . . . . . . 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 1143 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ)
2322rexrd 11311 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐹𝑧)) ∈ ℝ*)
2421simp2d 1144 . . . . 5 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ)
2524rexrd 11311 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐹𝑧)) ∈ ℝ*)
26 ovolfcl 25501 . . . . . . . 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 1143 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ)
2928rexrd 11311 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3029adantr 480 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1st ‘(𝐺𝐽)) ∈ ℝ*)
3127simp2d 1144 . . . . . 6 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ)
3231rexrd 11311 . . . . 5 ((𝜑𝐽 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
3332adantr 480 . . . 4 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2nd ‘(𝐺𝐽)) ∈ ℝ*)
34 iooin 13421 . . . 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 2777 . 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 13491 . 2 (if((1st ‘(𝐹𝑧)) ≤ (1st ‘(𝐺𝐽)), (1st ‘(𝐺𝐽)), (1st ‘(𝐹𝑧)))(,)if((2nd ‘(𝐹𝑧)) ≤ (2nd ‘(𝐺𝐽)), (2nd ‘(𝐹𝑧)), (2nd ‘(𝐺𝐽)))) ∈ ran (,)
3836, 37eqeltrdi 2849 1 (((𝜑𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝐽))) ∈ ran (,))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951  ifcif 4525  cop 4632   cuni 4907  Disj wdisj 5110   class class class wbr 5143   × cxp 5683  ran crn 5686  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  supcsup 9480  cr 11154  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296  cmin 11492  cn 12266  +crp 13034  (,)cioo 13387  seqcseq 14042  abscabs 15273  vol*covol 25497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-ioo 13391
This theorem is referenced by:  uniioombllem2  25618
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