Theorem ovolunnul 25876

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 1154	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
2		simp2 1155	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
3	1, 2	unssd 4193	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴 ∪ 𝐵) ⊆ ℝ)
4		ovolcl 25854	. . 3 ⊢ ((𝐴 ∪ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
5	3, 4	syl 18	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
6		ovolcl 25854	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
7	6	3ad2ant1 1151	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8		xrltnle 11415	. . . . 5 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
9	7, 5, 8	syl2anc 596	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
10	1	adantr 486	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → 𝐴 ⊆ ℝ)
11		mnfxr 11405	. . . . . . . . 9 ⊢ -∞ ∈ ℝ*
12	11	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → -∞ ∈ ℝ*)
13	10, 6	syl 18	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℝ*)
14	5	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
15		ovolge0 25857	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16	15	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17		ge0gtmnf 13342	. . . . . . . . . 10 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
18	7, 16, 17	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
19	18	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → -∞ < (vol*‘𝐴))
20		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)))
21		xrre2 13340	. . . . . . . 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 1406	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℝ)
23	2	adantr 486	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → 𝐵 ⊆ ℝ)
24		simpl3 1212	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) = 0)
25		0re 11349	. . . . . . . 8 ⊢ 0 ∈ ℝ
26	24, 25	eqeltrdi 2872	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) ∈ ℝ)
27		ovolun 25875	. . . . . . 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 852	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
29	24	oveq2d 7467	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
30	22	recnd 11376	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℂ)
31	30	addridd 11549	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
32	29, 31	eqtrd 2798	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = (vol*‘𝐴))
33	28, 32	breqtrd 5137	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴))
34	33	ex 418	. . . 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 128	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴))
37		ssun1 4179	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
38		ovolss 25861	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
39	37, 3, 38	sylancr 599	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
40	5, 7, 36, 39	xrletrid 13324	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  ℝcr 11238  0cc0 11239   + caddc 11242  -∞cmnf 11380  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383  vol*covol 25838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-q 13116  df-rp 13161  df-ioo 13520  df-ico 13522  df-fz 13680  df-fl 13972  df-seq 14185  df-exp 14245  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-ovol 25840

This theorem is used by:  mblfinlem2  38844