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Theorem ovolunlem2 25014
Description: Lemma for ovolun 25015. (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 13027 . . 3 (𝜑 → (𝐶 / 2) ∈ ℝ+)
6 eqid 2732 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
76ovolgelb 24996 . . 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 2732 . . . 4 seq1( + , ((abs ∘ − ) ∘ )) = seq1( + , ((abs ∘ − ) ∘ ))
1312ovolgelb 24996 . . 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 3226 . . 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 2732 . . . . . 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 2732 . . . . . 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 25013 . . . . 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 3210 . . 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 3070  cun 3946  cin 3947  wss 3948  ifcif 4528   cuni 4908   class class class wbr 5148  cmpt 5231   × cxp 5674  ran crn 5677  ccom 5680  cfv 6543  (class class class)co 7408  m cmap 8819  supcsup 9434  cr 11108  1c1 11110   + caddc 11112  *cxr 11246   < clt 11247  cle 11248  cmin 11443   / cdiv 11870  cn 12211  2c2 12266  +crp 12973  (,)cioo 13323  seqcseq 13965  abscabs 15180  vol*covol 24978
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12974  df-ioo 13327  df-ico 13329  df-fz 13484  df-fl 13756  df-seq 13966  df-exp 14027  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-ovol 24980
This theorem is referenced by:  ovolun  25015
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