MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolunlem2 Structured version   Visualization version   GIF version

Theorem ovolunlem2 24814
Description: Lemma for ovolun 24815. (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovolun.a (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
ovolun.b (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
ovolun.c (𝜑𝐶 ∈ ℝ+)
Assertion
Ref Expression
ovolunlem2 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))

Proof of Theorem ovolunlem2
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolun.a . . . 4 (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
21simpld 495 . . 3 (𝜑𝐴 ⊆ ℝ)
31simprd 496 . . 3 (𝜑 → (vol*‘𝐴) ∈ ℝ)
4 ovolun.c . . . 4 (𝜑𝐶 ∈ ℝ+)
54rphalfcld 12923 . . 3 (𝜑 → (𝐶 / 2) ∈ ℝ+)
6 eqid 2737 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
76ovolgelb 24796 . . 3 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ (𝐶 / 2) ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))))
82, 3, 5, 7syl3anc 1371 . 2 (𝜑 → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))))
9 ovolun.b . . . 4 (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
109simpld 495 . . 3 (𝜑𝐵 ⊆ ℝ)
119simprd 496 . . 3 (𝜑 → (vol*‘𝐵) ∈ ℝ)
12 eqid 2737 . . . 4 seq1( + , ((abs ∘ − ) ∘ )) = seq1( + , ((abs ∘ − ) ∘ ))
1312ovolgelb 24796 . . 3 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ ∧ (𝐶 / 2) ∈ ℝ+) → ∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))
1410, 11, 5, 13syl3anc 1371 . 2 (𝜑 → ∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))
15 reeanv 3215 . . 3 (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))) ↔ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ ∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))))
1613ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
1793ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
1843ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → 𝐶 ∈ ℝ+)
19 eqid 2737 . . . . . 6 seq1( + , ((abs ∘ − ) ∘ (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2)))))) = seq1( + , ((abs ∘ − ) ∘ (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2))))))
20 simp2l 1199 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
21 simp3ll 1244 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → 𝐴 ran ((,) ∘ 𝑔))
22 simp3lr 1245 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
23 simp2r 1200 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
24 simp3rl 1246 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → 𝐵 ran ((,) ∘ ))
25 simp3rr 1247 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
26 eqid 2737 . . . . . 6 (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2)))) = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2))))
2716, 17, 18, 6, 12, 19, 20, 21, 22, 23, 24, 25, 26ovolunlem1 24813 . . . . 5 ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ ((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))) → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
28273exp 1119 . . . 4 (𝜑 → ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → (((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))) → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))))
2928rexlimdvv 3202 . . 3 (𝜑 → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))) → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
3015, 29biimtrrid 242 . 2 (𝜑 → ((∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))) ∧ ∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))) → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
318, 14, 30mp2and 697 1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wcel 2106  wrex 3071  cun 3906  cin 3907  wss 3908  ifcif 4484   cuni 4863   class class class wbr 5103  cmpt 5186   × cxp 5629  ran crn 5632  ccom 5635  cfv 6493  (class class class)co 7351  m cmap 8723  supcsup 9334  cr 11008  1c1 11010   + caddc 11012  *cxr 11146   < clt 11147  cle 11148  cmin 11343   / cdiv 11770  cn 12111  2c2 12166  +crp 12869  (,)cioo 13218  seqcseq 13860  abscabs 15079  vol*covol 24778
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 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-er 8606  df-map 8725  df-en 8842  df-dom 8843  df-sdom 8844  df-sup 9336  df-inf 9337  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-div 11771  df-nn 12112  df-2 12174  df-3 12175  df-n0 12372  df-z 12458  df-uz 12722  df-rp 12870  df-ioo 13222  df-ico 13224  df-fz 13379  df-fl 13651  df-seq 13861  df-exp 13922  df-cj 14944  df-re 14945  df-im 14946  df-sqrt 15080  df-abs 15081  df-ovol 24780
This theorem is referenced by:  ovolun  24815
  Copyright terms: Public domain W3C validator