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Theorem ovolunnul 25549
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 1135 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
2 simp2 1136 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
31, 2unssd 4202 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴𝐵) ⊆ ℝ)
4 ovolcl 25527 . . 3 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
53, 4syl 17 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
6 ovolcl 25527 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
763ad2ant1 1132 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8 xrltnle 11326 . . . . 5 (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
97, 5, 8syl2anc 584 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
101adantr 480 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐴 ⊆ ℝ)
11 mnfxr 11316 . . . . . . . . 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 480 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
15 ovolge0 25530 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16153ad2ant1 1132 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17 ge0gtmnf 13211 . . . . . . . . . 10 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
187, 16, 17syl2anc 584 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
1918adantr 480 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → -∞ < (vol*‘𝐴))
20 simpr 484 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴𝐵)))
21 xrre2 13209 . . . . . . . 8 (((-∞ ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) ∧ (-∞ < (vol*‘𝐴) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵)))) → (vol*‘𝐴) ∈ ℝ)
2212, 13, 14, 19, 20, 21syl32anc 1377 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℝ)
232adantr 480 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐵 ⊆ ℝ)
24 simpl3 1192 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) = 0)
25 0re 11261 . . . . . . . 8 0 ∈ ℝ
2624, 25eqeltrdi 2847 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) ∈ ℝ)
27 ovolun 25548 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2810, 22, 23, 26, 27syl22anc 839 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2924oveq2d 7447 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
3022recnd 11287 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℂ)
3130addridd 11459 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
3229, 31eqtrd 2775 . . . . . 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 412 . . . 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 4188 . . 3 𝐴 ⊆ (𝐴𝐵)
38 ovolss 25534 . . 3 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
3937, 3, 38sylancr 587 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
405, 7, 36, 39xrletrid 13194 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cun 3961  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153   + caddc 11156  -∞cmnf 11291  *cxr 11292   < clt 11293  cle 11294  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-ioo 13388  df-ico 13390  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:  mblfinlem2  37645
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