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Theorem ovolunlem2 25247
Description: Lemma for ovolun 25248. (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 493 . . 3 (𝜑𝐴 ⊆ ℝ)
31simprd 494 . . 3 (𝜑 → (vol*‘𝐴) ∈ ℝ)
4 ovolun.c . . . 4 (𝜑𝐶 ∈ ℝ+)
54rphalfcld 13032 . . 3 (𝜑 → (𝐶 / 2) ∈ ℝ+)
6 eqid 2730 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
76ovolgelb 25229 . . 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 1369 . 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 493 . . 3 (𝜑𝐵 ⊆ ℝ)
119simprd 494 . . 3 (𝜑 → (vol*‘𝐵) ∈ ℝ)
12 eqid 2730 . . . 4 seq1( + , ((abs ∘ − ) ∘ )) = seq1( + , ((abs ∘ − ) ∘ ))
1312ovolgelb 25229 . . 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 1369 . 2 (𝜑 → ∃ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))
15 reeanv 3224 . . 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 1131 . . . . . 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 1131 . . . . . 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 1131 . . . . . 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 2730 . . . . . 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 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))))) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
21 simp3ll 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))))) → 𝐴 ran ((,) ∘ 𝑔))
22 simp3lr 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))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
23 simp2r 1198 . . . . . 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 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))))) → 𝐵 ran ((,) ∘ ))
25 simp3rr 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 ∘ − ) ∘ )), ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
26 eqid 2730 . . . . . 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 25246 . . . . 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 1117 . . . 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 3208 . . 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 695 1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085  wcel 2104  wrex 3068  cun 3945  cin 3946  wss 3947  ifcif 4527   cuni 4907   class class class wbr 5147  cmpt 5230   × cxp 5673  ran crn 5676  ccom 5679  cfv 6542  (class class class)co 7411  m cmap 8822  supcsup 9437  cr 11111  1c1 11113   + caddc 11115  *cxr 11251   < clt 11252  cle 11253  cmin 11448   / cdiv 11875  cn 12216  2c2 12271  +crp 12978  (,)cioo 13328  seqcseq 13970  abscabs 15185  vol*covol 25211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-ioo 13332  df-ico 13334  df-fz 13489  df-fl 13761  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-ovol 25213
This theorem is referenced by:  ovolun  25248
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