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Theorem ovolsplit 39974
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 4044 . . . . 5 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
21eqcomi 2630 . . . 4 𝐴 = ((𝐴𝐵) ∪ (𝐴𝐵))
32a1i 11 . . 3 (𝜑𝐴 = ((𝐴𝐵) ∪ (𝐴𝐵)))
43fveq2d 6193 . 2 (𝜑 → (vol*‘𝐴) = (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))))
5 ovolsplit.1 . . . . . . . . 9 (𝜑𝐴 ⊆ ℝ)
65ssinss1d 39040 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ ℝ)
75ssdifssd 3746 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ ℝ)
86, 7unssd 3787 . . . . . . 7 (𝜑 → ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ ℝ)
9 ovolcl 23240 . . . . . . 7 (((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ ℝ → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ*)
108, 9syl 17 . . . . . 6 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ*)
11 pnfge 11961 . . . . . 6 ((vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ∈ ℝ* → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
1210, 11syl 17 . . . . 5 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
1312adantr 481 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
14 oveq1 6654 . . . . . 6 ((vol*‘(𝐴𝐵)) = +∞ → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = (+∞ +𝑒 (vol*‘(𝐴𝐵))))
1514adantl 482 . . . . 5 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = (+∞ +𝑒 (vol*‘(𝐴𝐵))))
16 ovolcl 23240 . . . . . . . 8 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
177, 16syl 17 . . . . . . 7 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ*)
1817adantr 481 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
19 reex 10024 . . . . . . . . . . . . . 14 ℝ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℝ ∈ V)
2120, 5ssexd 4803 . . . . . . . . . . . 12 (𝜑𝐴 ∈ V)
22 difexg 4806 . . . . . . . . . . . 12 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
2321, 22syl 17 . . . . . . . . . . 11 (𝜑 → (𝐴𝐵) ∈ V)
24 elpwg 4164 . . . . . . . . . . 11 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 ℝ ↔ (𝐴𝐵) ⊆ ℝ))
2523, 24syl 17 . . . . . . . . . 10 (𝜑 → ((𝐴𝐵) ∈ 𝒫 ℝ ↔ (𝐴𝐵) ⊆ ℝ))
267, 25mpbird 247 . . . . . . . . 9 (𝜑 → (𝐴𝐵) ∈ 𝒫 ℝ)
27 ovolf 23244 . . . . . . . . . 10 vol*:𝒫 ℝ⟶(0[,]+∞)
2827ffvelrni 6356 . . . . . . . . 9 ((𝐴𝐵) ∈ 𝒫 ℝ → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
2926, 28syl 17 . . . . . . . 8 (𝜑 → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
3029xrge0nemnfd 39367 . . . . . . 7 (𝜑 → (vol*‘(𝐴𝐵)) ≠ -∞)
3130adantr 481 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ -∞)
32 xaddpnf2 12055 . . . . . 6 (((vol*‘(𝐴𝐵)) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ≠ -∞) → (+∞ +𝑒 (vol*‘(𝐴𝐵))) = +∞)
3318, 31, 32syl2anc 693 . . . . 5 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (+∞ +𝑒 (vol*‘(𝐴𝐵))) = +∞)
3415, 33eqtr2d 2656 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → +∞ = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
3513, 34breqtrd 4677 . . 3 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
36 simpl 473 . . . 4 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → 𝜑)
3720, 6sselpwd 4805 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ 𝒫 ℝ)
3827ffvelrni 6356 . . . . . . 7 ((𝐴𝐵) ∈ 𝒫 ℝ → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
3937, 38syl 17 . . . . . 6 (𝜑 → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
4039adantr 481 . . . . 5 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
41 neqne 2801 . . . . . 6 (¬ (vol*‘(𝐴𝐵)) = +∞ → (vol*‘(𝐴𝐵)) ≠ +∞)
4241adantl 482 . . . . 5 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ +∞)
43 ge0xrre 39567 . . . . 5 (((vol*‘(𝐴𝐵)) ∈ (0[,]+∞) ∧ (vol*‘(𝐴𝐵)) ≠ +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
4440, 42, 43syl2anc 693 . . . 4 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
4512adantr 481 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ +∞)
46 oveq2 6655 . . . . . . . . 9 ((vol*‘(𝐴𝐵)) = +∞ → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 +∞))
4746adantl 482 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 +∞))
48 ovolcl 23240 . . . . . . . . . . 11 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
496, 48syl 17 . . . . . . . . . 10 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ*)
5039xrge0nemnfd 39367 . . . . . . . . . 10 (𝜑 → (vol*‘(𝐴𝐵)) ≠ -∞)
51 xaddpnf1 12054 . . . . . . . . . 10 (((vol*‘(𝐴𝐵)) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ≠ -∞) → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5249, 50, 51syl2anc 693 . . . . . . . . 9 (𝜑 → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5352adantr 481 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → ((vol*‘(𝐴𝐵)) +𝑒 +∞) = +∞)
5447, 53eqtr2d 2656 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → +∞ = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
5545, 54breqtrd 4677 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
5655adantlr 751 . . . . 5 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
57 simpll 790 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → 𝜑)
58 simplr 792 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
5929adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ (0[,]+∞))
60 neqne 2801 . . . . . . . . 9 (¬ (vol*‘(𝐴𝐵)) = +∞ → (vol*‘(𝐴𝐵)) ≠ +∞)
6160adantl 482 . . . . . . . 8 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ≠ +∞)
62 ge0xrre 39567 . . . . . . . 8 (((vol*‘(𝐴𝐵)) ∈ (0[,]+∞) ∧ (vol*‘(𝐴𝐵)) ≠ +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6359, 61, 62syl2anc 693 . . . . . . 7 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6463adantlr 751 . . . . . 6 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6563ad2ant1 1081 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
66 simp2 1061 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
6773ad2ant1 1081 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
68 simp3 1062 . . . . . . . 8 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
69 ovolun 23261 . . . . . . . 8 ((((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
7065, 66, 67, 68, 69syl22anc 1326 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
71 rexadd 12060 . . . . . . . . 9 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))))
7271eqcomd 2627 . . . . . . . 8 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
73723adant1 1078 . . . . . . 7 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → ((vol*‘(𝐴𝐵)) + (vol*‘(𝐴𝐵))) = ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7470, 73breqtrd 4677 . . . . . 6 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7557, 58, 64, 74syl3anc 1325 . . . . 5 (((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7656, 75pm2.61dan 832 . . . 4 ((𝜑 ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7736, 44, 76syl2anc 693 . . 3 ((𝜑 ∧ ¬ (vol*‘(𝐴𝐵)) = +∞) → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
7835, 77pm2.61dan 832 . 2 (𝜑 → (vol*‘((𝐴𝐵) ∪ (𝐴𝐵))) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
794, 78eqbrtrd 4673 1 (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wne 2793  Vcvv 3198  cdif 3569  cun 3570  cin 3571  wss 3572  𝒫 cpw 4156   class class class wbr 4651  cfv 5886  (class class class)co 6647  cr 9932  0cc0 9933   + caddc 9936  +∞cpnf 10068  -∞cmnf 10069  *cxr 10070  cle 10072   +𝑒 cxad 11941  [,]cicc 12175  vol*covol 23225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-sup 8345  df-inf 8346  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-n0 11290  df-z 11375  df-uz 11685  df-q 11786  df-rp 11830  df-xadd 11944  df-ioo 12176  df-ico 12178  df-icc 12179  df-fz 12324  df-fl 12588  df-seq 12797  df-exp 12856  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-ovol 23227
This theorem is referenced by:  ismbl4  39979
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