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Theorem ovolunnul 25009
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 simp1 1137 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
2 simp2 1138 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
31, 2unssd 4186 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴𝐵) ⊆ ℝ)
4 ovolcl 24987 . . 3 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
53, 4syl 17 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
6 ovolcl 24987 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
763ad2ant1 1134 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8 xrltnle 11278 . . . . 5 (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
97, 5, 8syl2anc 585 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
101adantr 482 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐴 ⊆ ℝ)
11 mnfxr 11268 . . . . . . . . 9 -∞ ∈ ℝ*
1211a1i 11 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → -∞ ∈ ℝ*)
1310, 6syl 17 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℝ*)
145adantr 482 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
15 ovolge0 24990 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16153ad2ant1 1134 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17 ge0gtmnf 13148 . . . . . . . . . 10 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
187, 16, 17syl2anc 585 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
1918adantr 482 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → -∞ < (vol*‘𝐴))
20 simpr 486 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴𝐵)))
21 xrre2 13146 . . . . . . . 8 (((-∞ ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) ∧ (-∞ < (vol*‘𝐴) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵)))) → (vol*‘𝐴) ∈ ℝ)
2212, 13, 14, 19, 20, 21syl32anc 1379 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℝ)
232adantr 482 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐵 ⊆ ℝ)
24 simpl3 1194 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) = 0)
25 0re 11213 . . . . . . . 8 0 ∈ ℝ
2624, 25eqeltrdi 2842 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) ∈ ℝ)
27 ovolun 25008 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2810, 22, 23, 26, 27syl22anc 838 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2924oveq2d 7422 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
3022recnd 11239 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℂ)
3130addridd 11411 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
3229, 31eqtrd 2773 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = (vol*‘𝐴))
3328, 32breqtrd 5174 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴))
3433ex 414 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
359, 34sylbird 260 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
3635pm2.18d 127 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴))
37 ssun1 4172 . . 3 𝐴 ⊆ (𝐴𝐵)
38 ovolss 24994 . . 3 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
3937, 3, 38sylancr 588 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
405, 7, 36, 39xrletrid 13131 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cun 3946  wss 3948   class class class wbr 5148  cfv 6541  (class class class)co 7406  cr 11106  0cc0 11107   + caddc 11110  -∞cmnf 11243  *cxr 11244   < clt 11245  cle 11246  vol*covol 24971
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-inf 9435  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-ioo 13325  df-ico 13327  df-fz 13482  df-fl 13754  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-ovol 24973
This theorem is referenced by:  mblfinlem2  36515
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