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Theorem ovolsplit 44704
Description: The Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts, using addition for extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
ovolsplit.1 (𝜑𝐴 ⊆ ℝ)
Assertion
Ref Expression
ovolsplit (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))

Proof of Theorem ovolsplit
StepHypRef Expression
1 inundif 4479 . . . . 5 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
21eqcomi 2742 . . . 4 𝐴 = ((𝐴𝐵) ∪ (𝐴𝐵))
32a1i 11 . . 3 (𝜑𝐴 = ((𝐴𝐵) ∪ (𝐴𝐵)))
43fveq2d 6896 . 2 (𝜑 → (vol*‘𝐴) = (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))))
5 ovolsplit.1 . . . . . . . . 9 (𝜑𝐴 ⊆ ℝ)
65ssinss1d 43735 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ ℝ)
75ssdifssd 4143 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ ℝ)
86, 7unssd 4187 . . . . . . 7 (𝜑 → ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ ℝ)
9 ovolcl 24995 . . . . . . 7 (((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ ℝ → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ*)
108, 9syl 17 . . . . . 6 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ*)
11 pnfge 13110 . . . . . 6 ((vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ* → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
1210, 11syl 17 . . . . 5 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
1312adantr 482 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
14 oveq1 7416 . . . . . 6 ((vol*‘(𝐴𝐵)) = +∞ → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = (+∞ +𝑒 (vol*‘(𝐴𝐵))))
1514adantl 483 . . . . 5 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = (+∞ +𝑒 (vol*‘(𝐴𝐵))))
16 ovolcl 24995 . . . . . . . 8 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
177, 16syl 17 . . . . . . 7 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ*)
1817adantr 482 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
19 reex 11201 . . . . . . . . . . . . . 14 ℝ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℝ ∈ V)
2120, 5ssexd 5325 . . . . . . . . . . . 12 (𝜑𝐴 ∈ V)
2221difexd 5330 . . . . . . . . . . 11 (𝜑 → (𝐴𝐵) ∈ V)
23 elpwg 4606 . . . . . . . . . . 11 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 ℝ ↔ (𝐴𝐵) ⊆ ℝ))
2422, 23syl 17 . . . . . . . . . 10 (𝜑 → ((𝐴𝐵) ∈ 𝒫 ℝ ↔ (𝐴𝐵) ⊆ ℝ))
257, 24mpbird 257 . . . . . . . . 9 (𝜑 → (𝐴𝐵) ∈ 𝒫 ℝ)
26 ovolf 24999 . . . . . . . . . 10 vol*:𝒫 ℝ⟶(0[,]+∞)
2726ffvelcdmi 7086 . . . . . . . . 9 ((𝐴𝐵) ∈ 𝒫 ℝ → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
2825, 27syl 17 . . . . . . . 8 (𝜑 → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
2928xrge0nemnfd 44042 . . . . . . 7 (𝜑 → (vol*‘(𝐴𝐵)) ≠ -∞)
3029adantr 482 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ -∞)
31 xaddpnf2 13206 . . . . . 6 (((vol*‘(𝐴𝐵)) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ≠ -∞) → (+∞ +𝑒 (vol*‘(𝐴𝐵))) = +∞)
3218, 30, 31syl2anc 585 . . . . 5 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (+∞ +𝑒 (vol*‘(𝐴𝐵))) = +∞)
3315, 32eqtr2d 2774 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → +∞ = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
3413, 33breqtrd 5175 . . 3 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
35 simpl 484 . . . 4 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → 𝜑)
3620, 6sselpwd 5327 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ 𝒫 ℝ)
3726ffvelcdmi 7086 . . . . . . 7 ((𝐴𝐵) ∈ 𝒫 ℝ → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
3836, 37syl 17 . . . . . 6 (𝜑 → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
3938adantr 482 . . . . 5 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
40 neqne 2949 . . . . . 6 (¬ (vol*‘(𝐴𝐵)) = +∞ → (vol*‘(𝐴𝐵)) ≠ +∞)
4140adantl 483 . . . . 5 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ +∞)
42 ge0xrre 44244 . . . . 5 (((vol*‘(𝐴𝐵)) ∈ (0[,]+∞) ∧ (vol*‘(𝐴𝐵)) ≠ +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
4339, 41, 42syl2anc 585 . . . 4 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
4412adantr 482 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
45 oveq2 7417 . . . . . . . . 9 ((vol*‘(𝐴𝐵)) = +∞ → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 +∞))
4645adantl 483 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 +∞))
47 ovolcl 24995 . . . . . . . . . . 11 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
486, 47syl 17 . . . . . . . . . 10 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ*)
4938xrge0nemnfd 44042 . . . . . . . . . 10 (𝜑 → (vol*‘(𝐴𝐵)) ≠ -∞)
50 xaddpnf1 13205 . . . . . . . . . 10 (((vol*‘(𝐴𝐵)) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ≠ -∞) → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5148, 49, 50syl2anc 585 . . . . . . . . 9 (𝜑 → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5251adantr 482 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5346, 52eqtr2d 2774 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → +∞ = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
5444, 53breqtrd 5175 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
5554adantlr 714 . . . . 5 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
56 simpll 766 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → 𝜑)
57 simplr 768 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
5828adantr 482 . . . . . . . 8 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
59 neqne 2949 . . . . . . . . 9 (¬ (vol*‘(𝐴𝐵)) = +∞ → (vol*‘(𝐴𝐵)) ≠ +∞)
6059adantl 483 . . . . . . . 8 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ +∞)
61 ge0xrre 44244 . . . . . . . 8 (((vol*‘(𝐴𝐵)) ∈ (0[,]+∞) ∧ (vol*‘(𝐴𝐵)) ≠ +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6258, 60, 61syl2anc 585 . . . . . . 7 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6362adantlr 714 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6463ad2ant1 1134 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
65 simp2 1138 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6673ad2ant1 1134 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
67 simp3 1139 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
68 ovolun 25016 . . . . . . . 8 ((((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
6964, 65, 66, 67, 68syl22anc 838 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
70 rexadd 13211 . . . . . . . . 9 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
7170eqcomd 2739 . . . . . . . 8 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
72713adant1 1131 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7369, 72breqtrd 5175 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7456, 57, 63, 73syl3anc 1372 . . . . 5 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7555, 74pm2.61dan 812 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7635, 43, 75syl2anc 585 . . 3 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7734, 76pm2.61dan 812 . 2 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
784, 77eqbrtrd 5171 1 (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cdif 3946  cun 3947  cin 3948  wss 3949  𝒫 cpw 4603   class class class wbr 5149  cfv 6544  (class class class)co 7409  cr 11109  0cc0 11110   + caddc 11113  +∞cpnf 11245  -∞cmnf 11246  *cxr 11247  cle 11249   +𝑒 cxad 13090  [,]cicc 13327  vol*covol 24979
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xadd 13093  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fl 13757  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-ovol 24981
This theorem is referenced by:  ismbl4  44709
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