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Theorem ovolsplit 45944
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 4485 . . . . 5 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
21eqcomi 2744 . . . 4 𝐴 = ((𝐴𝐵) ∪ (𝐴𝐵))
32a1i 11 . . 3 (𝜑𝐴 = ((𝐴𝐵) ∪ (𝐴𝐵)))
43fveq2d 6911 . 2 (𝜑 → (vol*‘𝐴) = (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))))
5 ovolsplit.1 . . . . . . . . 9 (𝜑𝐴 ⊆ ℝ)
65ssinss1d 44988 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ ℝ)
75ssdifssd 4157 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ ℝ)
86, 7unssd 4202 . . . . . . 7 (𝜑 → ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ ℝ)
9 ovolcl 25527 . . . . . . 7 (((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ ℝ → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ*)
108, 9syl 17 . . . . . 6 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ*)
11 pnfge 13170 . . . . . 6 ((vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ* → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
1210, 11syl 17 . . . . 5 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
1312adantr 480 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
14 oveq1 7438 . . . . . 6 ((vol*‘(𝐴𝐵)) = +∞ → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = (+∞ +𝑒 (vol*‘(𝐴𝐵))))
1514adantl 481 . . . . 5 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = (+∞ +𝑒 (vol*‘(𝐴𝐵))))
16 ovolcl 25527 . . . . . . . 8 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
177, 16syl 17 . . . . . . 7 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ*)
1817adantr 480 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
19 reex 11244 . . . . . . . . . . . . . 14 ℝ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℝ ∈ V)
2120, 5ssexd 5330 . . . . . . . . . . . 12 (𝜑𝐴 ∈ V)
2221difexd 5337 . . . . . . . . . . 11 (𝜑 → (𝐴𝐵) ∈ V)
23 elpwg 4608 . . . . . . . . . . 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 25531 . . . . . . . . . 10 vol*:𝒫 ℝ⟶(0[,]+∞)
2726ffvelcdmi 7103 . . . . . . . . 9 ((𝐴𝐵) ∈ 𝒫 ℝ → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
2825, 27syl 17 . . . . . . . 8 (𝜑 → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
2928xrge0nemnfd 45282 . . . . . . 7 (𝜑 → (vol*‘(𝐴𝐵)) ≠ -∞)
3029adantr 480 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ -∞)
31 xaddpnf2 13266 . . . . . 6 (((vol*‘(𝐴𝐵)) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ≠ -∞) → (+∞ +𝑒 (vol*‘(𝐴𝐵))) = +∞)
3218, 30, 31syl2anc 584 . . . . 5 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (+∞ +𝑒 (vol*‘(𝐴𝐵))) = +∞)
3315, 32eqtr2d 2776 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → +∞ = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
3413, 33breqtrd 5174 . . 3 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
35 simpl 482 . . . 4 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → 𝜑)
3620, 6sselpwd 5334 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ 𝒫 ℝ)
3726ffvelcdmi 7103 . . . . . . 7 ((𝐴𝐵) ∈ 𝒫 ℝ → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
3836, 37syl 17 . . . . . 6 (𝜑 → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
3938adantr 480 . . . . 5 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
40 neqne 2946 . . . . . 6 (¬ (vol*‘(𝐴𝐵)) = +∞ → (vol*‘(𝐴𝐵)) ≠ +∞)
4140adantl 481 . . . . 5 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ +∞)
42 ge0xrre 45484 . . . . 5 (((vol*‘(𝐴𝐵)) ∈ (0[,]+∞) ∧ (vol*‘(𝐴𝐵)) ≠ +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
4339, 41, 42syl2anc 584 . . . 4 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
4412adantr 480 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
45 oveq2 7439 . . . . . . . . 9 ((vol*‘(𝐴𝐵)) = +∞ → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 +∞))
4645adantl 481 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 +∞))
47 ovolcl 25527 . . . . . . . . . . 11 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
486, 47syl 17 . . . . . . . . . 10 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ*)
4938xrge0nemnfd 45282 . . . . . . . . . 10 (𝜑 → (vol*‘(𝐴𝐵)) ≠ -∞)
50 xaddpnf1 13265 . . . . . . . . . 10 (((vol*‘(𝐴𝐵)) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ≠ -∞) → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5148, 49, 50syl2anc 584 . . . . . . . . 9 (𝜑 → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5251adantr 480 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5346, 52eqtr2d 2776 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → +∞ = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
5444, 53breqtrd 5174 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
5554adantlr 715 . . . . 5 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
56 simpll 767 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → 𝜑)
57 simplr 769 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
5828adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
59 neqne 2946 . . . . . . . . 9 (¬ (vol*‘(𝐴𝐵)) = +∞ → (vol*‘(𝐴𝐵)) ≠ +∞)
6059adantl 481 . . . . . . . 8 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ +∞)
61 ge0xrre 45484 . . . . . . . 8 (((vol*‘(𝐴𝐵)) ∈ (0[,]+∞) ∧ (vol*‘(𝐴𝐵)) ≠ +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6258, 60, 61syl2anc 584 . . . . . . 7 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6362adantlr 715 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6463ad2ant1 1132 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
65 simp2 1136 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6673ad2ant1 1132 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
67 simp3 1137 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
68 ovolun 25548 . . . . . . . 8 ((((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
6964, 65, 66, 67, 68syl22anc 839 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
70 rexadd 13271 . . . . . . . . 9 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
7170eqcomd 2741 . . . . . . . 8 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
72713adant1 1129 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7369, 72breqtrd 5174 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7456, 57, 63, 73syl3anc 1370 . . . . 5 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7555, 74pm2.61dan 813 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7635, 43, 75syl2anc 584 . . 3 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7734, 76pm2.61dan 813 . 2 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
784, 77eqbrtrd 5170 1 (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153   + caddc 11156  +∞cpnf 11290  -∞cmnf 11291  *cxr 11292  cle 11294   +𝑒 cxad 13150  [,]cicc 13387  vol*covol 25511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xadd 13153  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fl 13829  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-ovol 25513
This theorem is referenced by:  ismbl4  45949
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