Theorem ovolunnul 25659

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 4145	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴 ∪ 𝐵) ⊆ ℝ)
4		ovolcl 25637	. . 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 25637	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
7	6	3ad2ant1 1151	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8		xrltnle 11271	. . . . 5 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
9	7, 5, 8	syl2anc 595	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
10	1	adantr 485	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → 𝐴 ⊆ ℝ)
11		mnfxr 11261	. . . . . . . . 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 485	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*)
15		ovolge0 25640	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16	15	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17		ge0gtmnf 13193	. . . . . . . . . 10 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
18	7, 16, 17	syl2anc 595	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → -∞ < (vol*‘𝐴))
19	18	adantr 485	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → -∞ < (vol*‘𝐴))
20		simpr 489	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)))
21		xrre2 13191	. . . . . . . 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 1405	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℝ)
23	2	adantr 485	. . . . . . 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 11205	. . . . . . . 8 ⊢ 0 ∈ ℝ
26	24, 25	eqeltrdi 2871	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) ∈ ℝ)
27		ovolun 25658	. . . . . . 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 851	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
29	24	oveq2d 7426	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
30	22	recnd 11232	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℂ)
31	30	addridd 11405	. . . . . . 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 417	. . . 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 4131	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
38		ovolss 25644	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
39	37, 3, 38	sylancr 598	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
40	5, 7, 36, 39	xrletrid 13175	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∪ cun 3903   ⊆ wss 3905   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  ℝcr 11094  0cc0 11095   + caddc 11098  -∞cmnf 11236  ℝ^*cxr 11237   < clt 11238   ≤ cle 11239  vol*covol 25621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-inf 9399  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-q 12968  df-rp 13012  df-ioo 13371  df-ico 13373  df-fz 13531  df-fl 13821  df-seq 14034  df-exp 14094  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-ovol 25623

This theorem is referenced by:  mblfinlem2  38329