Theorem ovolunnul 25469

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 1137	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐴 ⊆ ℝ)
2		simp2 1138	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 𝐵 ⊆ ℝ)
3	1, 2	unssd 4146	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (𝐴 ∪ 𝐵) ⊆ ℝ)
4		ovolcl 25447	. . 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 25447	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
7	6	3ad2ant1 1134	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ∈ ℝ*)
8		xrltnle 11211	. . . . 5 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) → ((vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∪ 𝐵)) ≤ (vol*‘𝐴)))
9	7, 5, 8	syl2anc 585	. . . 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 11201	. . . . . . . . 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 25450	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
16	15	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → 0 ≤ (vol*‘𝐴))
17		ge0gtmnf 13099	. . . . . . . . . 10 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol*‘𝐴)) → -∞ < (vol*‘𝐴))
18	7, 16, 17	syl2anc 585	. . . . . . . . 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 13097	. . . . . . . 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 1381	. . . . . . 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 1195	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) = 0)
25		0re 11146	. . . . . . . 8 ⊢ 0 ∈ ℝ
26	24, 25	eqeltrdi 2845	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐵) ∈ ℝ)
27		ovolun 25468	. . . . . . 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 7384	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = ((vol*‘𝐴) + 0))
30	22	recnd 11172	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → (vol*‘𝐴) ∈ ℂ)
31	30	addridd 11345	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + 0) = (vol*‘𝐴))
32	29, 31	eqtrd 2772	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ∧ (vol*‘𝐴) < (vol*‘(𝐴 ∪ 𝐵))) → ((vol*‘𝐴) + (vol*‘𝐵)) = (vol*‘𝐴))
33	28, 32	breqtrd 5126	. . . . 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 4132	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
38		ovolss 25454	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
39	37, 3, 38	sylancr 588	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) ≤ (vol*‘(𝐴 ∪ 𝐵)))
40	5, 7, 36, 39	xrletrid 13081	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∪ cun 3901   ⊆ wss 3903   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  ℝcr 11037  0cc0 11038   + caddc 11041  -∞cmnf 11176  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179  vol*covol 25431

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-ioo 13277  df-ico 13279  df-fz 13436  df-fl 13724  df-seq 13937  df-exp 13997  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-ovol 25433

This theorem is referenced by:  mblfinlem2  37909