Theorem ovolunnul 24369

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 1138	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
2		simp2 1139	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
3	1, 2	unssd 4090	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴 ∪ 𝐵) ⊆ ℝ)
4		ovolcl 24347	. . 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 24347	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
7	6	3ad2ant1 1135	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8		xrltnle 10883	. . . . 5 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
9	7, 5, 8	syl2anc 587	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
10	1	adantr 484	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → 𝐴 ⊆ ℝ)
11		mnfxr 10873	. . . . . . . . 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 484	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
15		ovolge0 24350	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16	15	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17		ge0gtmnf 12745	. . . . . . . . . 10 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
18	7, 16, 17	syl2anc 587	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
19	18	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → -∞ < (vol*‘𝐴))
20		simpr 488	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)))
21		xrre2 12743	. . . . . . . 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 484	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → 𝐵 ⊆ ℝ)
24		simpl3 1195	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) = 0)
25		0re 10818	. . . . . . . 8 ⊢ 0 ∈ ℝ
26	24, 25	eqeltrdi 2842	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) ∈ ℝ)
27		ovolun 24368	. . . . . . 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 839	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
29	24	oveq2d 7218	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
30	22	recnd 10844	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℂ)
31	30	addid1d 11015	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
32	29, 31	eqtrd 2774	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = (vol*‘𝐴))
33	28, 32	breqtrd 5069	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴))
34	33	ex 416	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
35	9, 34	sylbird 263	. . 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 4076	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
38		ovolss 24354	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
39	37, 3, 38	sylancr 590	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
40	5, 7, 36, 39	xrletrid 12728	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ∪ cun 3855   ⊆ wss 3857   class class class wbr 5043  ‘cfv 6369  (class class class)co 7202  ℝcr 10711  0cc0 10712   + caddc 10715  -∞cmnf 10848  ℝ^*cxr 10849   < clt 10850   ≤ cle 10851  vol*covol 24331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-sup 9047  df-inf 9048  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-n0 12074  df-z 12160  df-uz 12422  df-q 12528  df-rp 12570  df-ioo 12922  df-ico 12924  df-fz 13079  df-fl 13350  df-seq 13558  df-exp 13619  df-cj 14645  df-re 14646  df-im 14647  df-sqrt 14781  df-abs 14782  df-ovol 24333

This theorem is referenced by:  mblfinlem2  35509