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Theorem ovolunnul 23262
Description: Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)
Assertion
Ref Expression
ovolunnul ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) = (vol*‘𝐴))

Proof of Theorem ovolunnul
StepHypRef Expression
1 ovolcl 23240 . . . . . 6 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
213ad2ant1 1081 . . . . 5 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
3 simp1 1060 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
4 simp2 1061 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
53, 4unssd 3787 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴𝐵) ⊆ ℝ)
6 ovolcl 23240 . . . . . 6 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
75, 6syl 17 . . . . 5 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
8 xrltnle 10102 . . . . 5 (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
92, 7, 8syl2anc 693 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
103adantr 481 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐴 ⊆ ℝ)
11 mnfxr 10093 . . . . . . . . 9 -∞ ∈ ℝ*
1211a1i 11 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → -∞ ∈ ℝ*)
1310, 1syl 17 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℝ*)
147adantr 481 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
15 ovolge0 23243 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16153ad2ant1 1081 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17 ge0gtmnf 12000 . . . . . . . . . 10 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
182, 16, 17syl2anc 693 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
1918adantr 481 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → -∞ < (vol*‘𝐴))
20 simpr 477 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴𝐵)))
21 xrre2 11998 . . . . . . . 8 (((-∞ ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) ∧ (-∞ < (vol*‘𝐴) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵)))) → (vol*‘𝐴) ∈ ℝ)
2212, 13, 14, 19, 20, 21syl32anc 1333 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℝ)
234adantr 481 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐵 ⊆ ℝ)
24 simpl3 1065 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) = 0)
25 0re 10037 . . . . . . . 8 0 ∈ ℝ
2624, 25syl6eqel 2708 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) ∈ ℝ)
27 ovolun 23261 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2810, 22, 23, 26, 27syl22anc 1326 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2924oveq2d 6663 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
3022recnd 10065 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℂ)
3130addid1d 10233 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
3229, 31eqtrd 2655 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = (vol*‘𝐴))
3328, 32breqtrd 4677 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴))
3433ex 450 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
359, 34sylbird 250 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
3635pm2.18d 124 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴))
37 ssun1 3774 . . 3 𝐴 ⊆ (𝐴𝐵)
38 ovolss 23247 . . 3 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
3937, 5, 38sylancr 695 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
40 xrletri3 11982 . . 3 (((vol*‘(𝐴𝐵)) ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘(𝐴𝐵)) = (vol*‘𝐴) ↔ ((vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴) ∧ (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))))
417, 2, 40syl2anc 693 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘(𝐴𝐵)) = (vol*‘𝐴) ↔ ((vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴) ∧ (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))))
4236, 39, 41mpbir2and 957 1 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) = (vol*‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  cun 3570  wss 3572   class class class wbr 4651  cfv 5886  (class class class)co 6647  cr 9932  0cc0 9933   + caddc 9936  -∞cmnf 10069  *cxr 10070   < clt 10071  cle 10072  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-ioo 12176  df-ico 12178  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:  mblfinlem2  33427
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