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

Theorem ovolunlem2 24885
Description: Lemma for ovolun 24886. (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 496 . . 3 (𝜑𝐴 ⊆ ℝ)
31simprd 497 . . 3 (𝜑 → (vol*‘𝐴) ∈ ℝ)
4 ovolun.c . . . 4 (𝜑𝐶 ∈ ℝ+)
54rphalfcld 12977 . . 3 (𝜑 → (𝐶 / 2) ∈ ℝ+)
6 eqid 2733 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
76ovolgelb 24867 . . 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 1372 . 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 496 . . 3 (𝜑𝐵 ⊆ ℝ)
119simprd 497 . . 3 (𝜑 → (vol*‘𝐵) ∈ ℝ)
12 eqid 2733 . . . 4 seq1( + , ((abs ∘ − ) ∘ )) = seq1( + , ((abs ∘ − ) ∘ ))
1312ovolgelb 24867 . . 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 1372 . 2 (𝜑 → ∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))
15 reeanv 3216 . . 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 1134 . . . . . 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 1134 . . . . . 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 1134 . . . . . 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 2733 . . . . . 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 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))))) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
21 simp3ll 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))))) → 𝐴 ran ((,) ∘ 𝑔))
22 simp3lr 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))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
23 simp2r 1201 . . . . . 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 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))))) → 𝐵 ran ((,) ∘ ))
25 simp3rr 1248 . . . . . 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 2733 . . . . . 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 24884 . . . . 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 1120 . . . 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 3201 . . 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 698 1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wrex 3070  cun 3912  cin 3913  wss 3914  ifcif 4490   cuni 4869   class class class wbr 5109  cmpt 5192   × cxp 5635  ran crn 5638  ccom 5641  cfv 6500  (class class class)co 7361  m cmap 8771  supcsup 9384  cr 11058  1c1 11060   + caddc 11062  *cxr 11196   < clt 11197  cle 11198  cmin 11393   / cdiv 11820  cn 12161  2c2 12216  +crp 12923  (,)cioo 13273  seqcseq 13915  abscabs 15128  vol*covol 24849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-ioo 13277  df-ico 13279  df-fz 13434  df-fl 13706  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-ovol 24851
This theorem is referenced by:  ovolun  24886
  Copyright terms: Public domain W3C validator