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Theorem ovolunnul 24107
 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 1133 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
2 simp2 1134 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
31, 2unssd 4116 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴𝐵) ⊆ ℝ)
4 ovolcl 24085 . . 3 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
53, 4syl 17 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
6 ovolcl 24085 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
763ad2ant1 1130 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8 xrltnle 10701 . . . . 5 (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
97, 5, 8syl2anc 587 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
101adantr 484 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐴 ⊆ ℝ)
11 mnfxr 10691 . . . . . . . . 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 484 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ∈ ℝ*)
15 ovolge0 24088 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16153ad2ant1 1130 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17 ge0gtmnf 12557 . . . . . . . . . 10 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
187, 16, 17syl2anc 587 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
1918adantr 484 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → -∞ < (vol*‘𝐴))
20 simpr 488 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴𝐵)))
21 xrre2 12555 . . . . . . . 8 (((-∞ ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) ∧ (-∞ < (vol*‘𝐴) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵)))) → (vol*‘𝐴) ∈ ℝ)
2212, 13, 14, 19, 20, 21syl32anc 1375 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℝ)
232adantr 484 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → 𝐵 ⊆ ℝ)
24 simpl3 1190 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) = 0)
25 0re 10636 . . . . . . . 8 0 ∈ ℝ
2624, 25eqeltrdi 2901 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐵) ∈ ℝ)
27 ovolun 24106 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2810, 22, 23, 26, 27syl22anc 837 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
2924oveq2d 7155 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
3022recnd 10662 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘𝐴) ∈ ℂ)
3130addid1d 10833 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
3229, 31eqtrd 2836 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = (vol*‘𝐴))
3328, 32breqtrd 5059 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴𝐵))) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴))
3433ex 416 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴𝐵)) → (vol*‘(𝐴𝐵)) ≤ (vol*‘𝐴)))
359, 34sylbird 263 . . 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 4102 . . 3 𝐴 ⊆ (𝐴𝐵)
38 ovolss 24092 . . 3 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
3937, 3, 38sylancr 590 . 2 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴𝐵)))
405, 7, 36, 39xrletrid 12540 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ∪ cun 3882   ⊆ wss 3884   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  ℝcr 10529  0cc0 10530   + caddc 10533  -∞cmnf 10666  ℝ*cxr 10667   < clt 10668   ≤ cle 10669  vol*covol 24069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-inf 8895  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-ioo 12734  df-ico 12736  df-fz 12890  df-fl 13161  df-seq 13369  df-exp 13430  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-ovol 24071 This theorem is referenced by:  mblfinlem2  35088
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