Theorem ovolunnul 25399

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*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴))

Proof of Theorem ovolunnul

Step	Hyp	Ref	Expression
1		simp1 1136	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
2		simp2 1137	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
3	1, 2	unssd 4143	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴 ∪ 𝐵) ⊆ ℝ)
4		ovolcl 25377	. . 3 ⊢ ((𝐴 ∪ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
5	3, 4	syl 17	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
6		ovolcl 25377	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
7	6	3ad2ant1 1133	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8		xrltnle 11182	. . . . 5 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
9	7, 5, 8	syl2anc 584	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
10	1	adantr 480	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → 𝐴 ⊆ ℝ)
11		mnfxr 11172	. . . . . . . . 9 ⊢ -∞ ∈ ℝ*
12	11	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → -∞ ∈ ℝ*)
13	10, 6	syl 17	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℝ*)
14	5	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
15		ovolge0 25380	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16	15	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17		ge0gtmnf 13074	. . . . . . . . . 10 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
18	7, 16, 17	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
19	18	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → -∞ < (vol*‘𝐴))
20		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)))
21		xrre2 13072	. . . . . . . 8 ⊢ (((-∞ ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) ∧ (-∞ < (vol*‘𝐴) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)))) → (vol*‘𝐴) ∈ ℝ)
22	12, 13, 14, 19, 20, 21	syl32anc 1380	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℝ)
23	2	adantr 480	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → 𝐵 ⊆ ℝ)
24		simpl3 1194	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) = 0)
25		0re 11117	. . . . . . . 8 ⊢ 0 ∈ ℝ
26	24, 25	eqeltrdi 2836	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) ∈ ℝ)
27		ovolun 25398	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
28	10, 22, 23, 26, 27	syl22anc 838	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
29	24	oveq2d 7365	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
30	22	recnd 11143	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℂ)
31	30	addridd 11316	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
32	29, 31	eqtrd 2764	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = (vol*‘𝐴))
33	28, 32	breqtrd 5118	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴))
34	33	ex 412	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
35	9, 34	sylbird 260	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
36	35	pm2.18d 127	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴))
37		ssun1 4129	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
38		ovolss 25384	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
39	37, 3, 38	sylancr 587	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
40	5, 7, 36, 39	xrletrid 13057	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3901   ⊆ wss 3903   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  ℝcr 11008  0cc0 11009   + caddc 11012  -∞cmnf 11147  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150  vol*covol 25361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-ioo 13252  df-ico 13254  df-fz 13411  df-fl 13696  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-ovol 25363

This theorem is referenced by:  mblfinlem2  37642