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Theorem ovollb2 23064
Description: It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 23054). (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypothesis
Ref Expression
ovollb2.1 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
Assertion
Ref Expression
ovollb2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Proof of Theorem ovollb2
Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝐴 ran ([,] ∘ 𝐹))
2 ovolficcss 23045 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
32adantr 480 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
41, 3sstrd 3578 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ℝ)
5 ovolcl 23053 . . . . . 6 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
64, 5syl 17 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ∈ ℝ*)
76adantr 480 . . . 4 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ∈ ℝ*)
8 pnfge 11840 . . . 4 ((vol*‘𝐴) ∈ ℝ* → (vol*‘𝐴) ≤ +∞)
97, 8syl 17 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ≤ +∞)
10 simpr 476 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → sup(ran 𝑆, ℝ*, < ) = +∞)
119, 10breqtrrd 4611 . 2 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
12 eqid 2610 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
13 ovollb2.1 . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
1412, 13ovolsf 23048 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1514adantr 480 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝑆:ℕ⟶(0[,)+∞))
16 frn 5966 . . . . . . 7 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
1715, 16syl 17 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ (0[,)+∞))
18 rge0ssre 12151 . . . . . 6 (0[,)+∞) ⊆ ℝ
1917, 18syl6ss 3580 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ ℝ)
20 1nn 10908 . . . . . . . 8 1 ∈ ℕ
21 fdm 5964 . . . . . . . . 9 (𝑆:ℕ⟶(0[,)+∞) → dom 𝑆 = ℕ)
2215, 21syl 17 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → dom 𝑆 = ℕ)
2320, 22syl5eleqr 2695 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 1 ∈ dom 𝑆)
24 ne0i 3880 . . . . . . 7 (1 ∈ dom 𝑆 → dom 𝑆 ≠ ∅)
2523, 24syl 17 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → dom 𝑆 ≠ ∅)
26 dm0rn0 5263 . . . . . . 7 (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
2726necon3bii 2834 . . . . . 6 (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
2825, 27sylib 207 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ≠ ∅)
29 supxrre2 12033 . . . . 5 ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅) → (sup(ran 𝑆, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ*, < ) ≠ +∞))
3019, 28, 29syl2anc 691 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (sup(ran 𝑆, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ*, < ) ≠ +∞))
3130biimpar 501 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
32 fveq2 6103 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3332fveq2d 6107 . . . . . . . . 9 (𝑚 = 𝑛 → (1st ‘(𝐹𝑚)) = (1st ‘(𝐹𝑛)))
34 oveq2 6557 . . . . . . . . . 10 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3534oveq2d 6565 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝑥 / 2) / (2↑𝑚)) = ((𝑥 / 2) / (2↑𝑛)))
3633, 35oveq12d 6567 . . . . . . . 8 (𝑚 = 𝑛 → ((1st ‘(𝐹𝑚)) − ((𝑥 / 2) / (2↑𝑚))) = ((1st ‘(𝐹𝑛)) − ((𝑥 / 2) / (2↑𝑛))))
3732fveq2d 6107 . . . . . . . . 9 (𝑚 = 𝑛 → (2nd ‘(𝐹𝑚)) = (2nd ‘(𝐹𝑛)))
3837, 35oveq12d 6567 . . . . . . . 8 (𝑚 = 𝑛 → ((2nd ‘(𝐹𝑚)) + ((𝑥 / 2) / (2↑𝑚))) = ((2nd ‘(𝐹𝑛)) + ((𝑥 / 2) / (2↑𝑛))))
3936, 38opeq12d 4348 . . . . . . 7 (𝑚 = 𝑛 → ⟨((1st ‘(𝐹𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩ = ⟨((1st ‘(𝐹𝑛)) − ((𝑥 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝑥 / 2) / (2↑𝑛)))⟩)
4039cbvmptv 4678 . . . . . 6 (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝑥 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝑥 / 2) / (2↑𝑛)))⟩)
41 eqid 2610 . . . . . 6 seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩))) = seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩)))
42 simplll 794 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
43 simpllr 795 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐴 ran ([,] ∘ 𝐹))
44 simpr 476 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
45 simplr 788 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
4613, 40, 41, 42, 43, 44, 45ovollb2lem 23063 . . . . 5 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
4746ralrimiva 2949 . . . 4 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
48 xralrple 11910 . . . . 5 (((vol*‘𝐴) ∈ ℝ* ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ((vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥)))
496, 48sylan 487 . . . 4 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ((vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥)))
5047, 49mpbird 246 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
5131, 50syldan 486 . 2 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
5211, 51pm2.61dane 2869 1 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  cin 3539  wss 3540  c0 3874  cop 4131   cuni 4372   class class class wbr 4583  cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039  ccom 5042  wf 5800  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  supcsup 8229  cr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950  *cxr 9952   < clt 9953  cle 9954  cmin 10145   / cdiv 10563  cn 10897  2c2 10947  +crp 11708  [,)cico 12048  [,]cicc 12049  seqcseq 12663  cexp 12722  abscabs 13822  vol*covol 23038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-ovol 23040
This theorem is referenced by:  ovolctb  23065  ovolicc1  23091  ioombl1lem4  23136  uniiccvol  23154
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