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Theorem ovollb2 25436
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 25426). (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 483 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝐴 ran ([,] ∘ 𝐹))
2 ovolficcss 25416 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
32adantr 479 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
41, 3sstrd 3990 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ℝ)
5 ovolcl 25425 . . . . . 6 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
64, 5syl 17 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ∈ ℝ*)
76adantr 479 . . . 4 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ∈ ℝ*)
8 pnfge 13148 . . . 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 483 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → sup(ran 𝑆, ℝ*, < ) = +∞)
119, 10breqtrrd 5178 . 2 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
12 eqid 2727 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
13 ovollb2.1 . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
1412, 13ovolsf 25419 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1514adantr 479 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 𝑆:ℕ⟶(0[,)+∞))
1615frnd 6733 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ (0[,)+∞))
17 rge0ssre 13471 . . . . . 6 (0[,)+∞) ⊆ ℝ
1816, 17sstrdi 3992 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ ℝ)
19 1nn 12259 . . . . . . . 8 1 ∈ ℕ
2015fdmd 6736 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → dom 𝑆 = ℕ)
2119, 20eleqtrrid 2835 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → 1 ∈ dom 𝑆)
2221ne0d 4337 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → dom 𝑆 ≠ ∅)
23 dm0rn0 5929 . . . . . . 7 (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
2423necon3bii 2989 . . . . . 6 (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
2522, 24sylib 217 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → ran 𝑆 ≠ ∅)
26 supxrre2 13348 . . . . 5 ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅) → (sup(ran 𝑆, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ*, < ) ≠ +∞))
2718, 25, 26syl2anc 582 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (sup(ran 𝑆, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ*, < ) ≠ +∞))
2827biimpar 476 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
29 2fveq3 6905 . . . . . . . . 9 (𝑚 = 𝑛 → (1st ‘(𝐹𝑚)) = (1st ‘(𝐹𝑛)))
30 oveq2 7432 . . . . . . . . . 10 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3130oveq2d 7440 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝑥 / 2) / (2↑𝑚)) = ((𝑥 / 2) / (2↑𝑛)))
3229, 31oveq12d 7442 . . . . . . . 8 (𝑚 = 𝑛 → ((1st ‘(𝐹𝑚)) − ((𝑥 / 2) / (2↑𝑚))) = ((1st ‘(𝐹𝑛)) − ((𝑥 / 2) / (2↑𝑛))))
33 2fveq3 6905 . . . . . . . . 9 (𝑚 = 𝑛 → (2nd ‘(𝐹𝑚)) = (2nd ‘(𝐹𝑛)))
3433, 31oveq12d 7442 . . . . . . . 8 (𝑚 = 𝑛 → ((2nd ‘(𝐹𝑚)) + ((𝑥 / 2) / (2↑𝑚))) = ((2nd ‘(𝐹𝑛)) + ((𝑥 / 2) / (2↑𝑛))))
3532, 34opeq12d 4884 . . . . . . 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 5263 . . . . . 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 2727 . . . . . 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 773 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
39 simpllr 774 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐴 ran ([,] ∘ 𝐹))
40 simpr 483 . . . . . 6 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
41 simplr 767 . . . . . 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 25435 . . . . 5 ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
4342ralrimiva 3142 . . . 4 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
44 xralrple 13222 . . . . 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 578 . . . 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 256 . . 3 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
4728, 46syldan 589 . 2 (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
4811, 47pm2.61dane 3025 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 205  wa 394   = wceq 1533  wcel 2098  wne 2936  wral 3057  cin 3946  wss 3947  c0 4324  cop 4636   cuni 4910   class class class wbr 5150  cmpt 5233   × cxp 5678  dom cdm 5680  ran crn 5681  ccom 5684  wf 6547  cfv 6551  (class class class)co 7424  1st c1st 7995  2nd c2nd 7996  supcsup 9469  cr 11143  0cc0 11144  1c1 11145   + caddc 11147  +∞cpnf 11281  *cxr 11283   < clt 11284  cle 11285  cmin 11480   / cdiv 11907  cn 12248  2c2 12303  +crp 13012  [,)cico 13364  [,]cicc 13365  seqcseq 14004  cexp 14064  abscabs 15219  vol*covol 25409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-inf2 9670  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221  ax-pre-sup 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-se 5636  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-isom 6560  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-er 8729  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-sup 9471  df-inf 9472  df-oi 9539  df-card 9968  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-div 11908  df-nn 12249  df-2 12311  df-3 12312  df-n0 12509  df-z 12595  df-uz 12859  df-q 12969  df-rp 13013  df-ioo 13366  df-ico 13368  df-icc 13369  df-fz 13523  df-fzo 13666  df-seq 14005  df-exp 14065  df-hash 14328  df-cj 15084  df-re 15085  df-im 15086  df-sqrt 15220  df-abs 15221  df-clim 15470  df-sum 15671  df-ovol 25411
This theorem is referenced by:  ovolctb  25437  ovolicc1  25463  ioombl1lem4  25508  uniiccvol  25527
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