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Theorem ovollb2 24102
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 24092). (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 488 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝐴 ran ([,] ∘ 𝐹))
2 ovolficcss 24082 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
32adantr 484 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
41, 3sstrd 3963 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ℝ)
5 ovolcl 24091 . . . . . 6 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
64, 5syl 17 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ∈ ℝ*)
76adantr 484 . . . 4 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ∈ ℝ*)
8 pnfge 12524 . . . 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 488 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → sup(ran 𝑆, ℝ*, < ) = +∞)
119, 10breqtrrd 5081 . 2 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
12 eqid 2824 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
13 ovollb2.1 . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
1412, 13ovolsf 24085 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1514adantr 484 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝑆:ℕ⟶(0[,)+∞))
1615frnd 6512 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ (0[,)+∞))
17 rge0ssre 12845 . . . . . 6 (0[,)+∞) ⊆ ℝ
1816, 17sstrdi 3965 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ ℝ)
19 1nn 11647 . . . . . . . 8 1 ∈ ℕ
2015fdmd 6515 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → dom 𝑆 = ℕ)
2119, 20eleqtrrid 2923 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 1 ∈ dom 𝑆)
2221ne0d 4284 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → dom 𝑆 ≠ ∅)
23 dm0rn0 5783 . . . . . . 7 (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
2423necon3bii 3066 . . . . . 6 (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
2522, 24sylib 221 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ≠ ∅)
26 supxrre2 12723 . . . . 5 ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅) → (sup(ran 𝑆, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ*, < ) ≠ +∞))
2718, 25, 26syl2anc 587 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (sup(ran 𝑆, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ*, < ) ≠ +∞))
2827biimpar 481 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
29 2fveq3 6668 . . . . . . . . 9 (𝑚 = 𝑛 → (1st ‘(𝐹𝑚)) = (1st ‘(𝐹𝑛)))
30 oveq2 7159 . . . . . . . . . 10 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3130oveq2d 7167 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝑥 / 2) / (2↑𝑚)) = ((𝑥 / 2) / (2↑𝑛)))
3229, 31oveq12d 7169 . . . . . . . 8 (𝑚 = 𝑛 → ((1st ‘(𝐹𝑚)) − ((𝑥 / 2) / (2↑𝑚))) = ((1st ‘(𝐹𝑛)) − ((𝑥 / 2) / (2↑𝑛))))
33 2fveq3 6668 . . . . . . . . 9 (𝑚 = 𝑛 → (2nd ‘(𝐹𝑚)) = (2nd ‘(𝐹𝑛)))
3433, 31oveq12d 7169 . . . . . . . 8 (𝑚 = 𝑛 → ((2nd ‘(𝐹𝑚)) + ((𝑥 / 2) / (2↑𝑚))) = ((2nd ‘(𝐹𝑛)) + ((𝑥 / 2) / (2↑𝑛))))
3532, 34opeq12d 4797 . . . . . . 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)))⟩)
3635cbvmptv 5156 . . . . . 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)))⟩)
37 eqid 2824 . . . . . 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)))⟩)))
38 simplll 774 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
39 simpllr 775 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐴 ran ([,] ∘ 𝐹))
40 simpr 488 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
41 simplr 768 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
4213, 36, 37, 38, 39, 40, 41ovollb2lem 24101 . . . . 5 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
4342ralrimiva 3177 . . . 4 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
44 xralrple 12597 . . . . 5 (((vol*‘𝐴) ∈ ℝ* ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ((vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥)))
456, 44sylan 583 . . . 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)))
4643, 45mpbird 260 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
4728, 46syldan 594 . 2 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
4811, 47pm2.61dane 3101 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 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  cin 3918  wss 3919  c0 4276  cop 4556   cuni 4824   class class class wbr 5053  cmpt 5133   × cxp 5541  dom cdm 5543  ran crn 5544  ccom 5547  wf 6341  cfv 6345  (class class class)co 7151  1st c1st 7684  2nd c2nd 7685  supcsup 8903  cr 10536  0cc0 10537  1c1 10538   + caddc 10540  +∞cpnf 10672  *cxr 10674   < clt 10675  cle 10676  cmin 10870   / cdiv 11297  cn 11636  2c2 11691  +crp 12388  [,)cico 12739  [,]cicc 12740  seqcseq 13375  cexp 13436  abscabs 14595  vol*covol 24075
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-inf2 9103  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-isom 6354  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-sup 8905  df-inf 8906  df-oi 8973  df-card 9367  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11637  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-ioo 12741  df-ico 12743  df-icc 12744  df-fz 12897  df-fzo 13040  df-seq 13376  df-exp 13437  df-hash 13698  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-ovol 24077
This theorem is referenced by:  ovolctb  24103  ovolicc1  24129  ioombl1lem4  24174  uniiccvol  24193
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