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Theorem ovolunlem2 24100
Description: Lemma for ovolun 24101. (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 498 . . 3 (𝜑𝐴 ⊆ ℝ)
31simprd 499 . . 3 (𝜑 → (vol*‘𝐴) ∈ ℝ)
4 ovolun.c . . . 4 (𝜑𝐶 ∈ ℝ+)
54rphalfcld 12431 . . 3 (𝜑 → (𝐶 / 2) ∈ ℝ+)
6 eqid 2822 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
76ovolgelb 24082 . . 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 1368 . 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 498 . . 3 (𝜑𝐵 ⊆ ℝ)
119simprd 499 . . 3 (𝜑 → (vol*‘𝐵) ∈ ℝ)
12 eqid 2822 . . . 4 seq1( + , ((abs ∘ − ) ∘ )) = seq1( + , ((abs ∘ − ) ∘ ))
1312ovolgelb 24082 . . 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 1368 . 2 (𝜑 → ∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))
15 reeanv 3348 . . 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 1130 . . . . . 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 1130 . . . . . 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 1130 . . . . . 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 2822 . . . . . 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 1196 . . . . . 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 1241 . . . . . 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 1242 . . . . . 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 1197 . . . . . 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 1243 . . . . . 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 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))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
26 eqid 2822 . . . . . 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 24099 . . . . 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 1116 . . . 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 3279 . . 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, 29syl5bir 246 . 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 399  w3a 1084  wcel 2114  wrex 3131  cun 3906  cin 3907  wss 3908  ifcif 4439   cuni 4813   class class class wbr 5042  cmpt 5122   × cxp 5530  ran crn 5533  ccom 5536  cfv 6334  (class class class)co 7140  m cmap 8393  supcsup 8892  cr 10525  1c1 10527   + caddc 10529  *cxr 10663   < clt 10664  cle 10665  cmin 10859   / cdiv 11286  cn 11625  2c2 11680  +crp 12377  (,)cioo 12726  seqcseq 13364  abscabs 14584  vol*covol 24064
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-inf 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ioo 12730  df-ico 12732  df-fz 12886  df-fl 13157  df-seq 13365  df-exp 13426  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-ovol 24066
This theorem is referenced by:  ovolun  24101
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