Theorem ismbl2 25455

Description: From ovolun 25427, it suffices to show that the measure of 𝑥 is at least the sum of the measures of 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)

Assertion

Ref	Expression
ismbl2	⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl2

Step	Hyp	Ref	Expression
1		ismbl 25454	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))
2		elpwi 4554	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3		inundif 4426	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ∪ (𝑥 ∖ 𝐴)) = 𝑥
4	3	fveq2i 6825	. . . . . . . . 9 ⊢ (vol*‘((𝑥 ∩ 𝐴) ∪ (𝑥 ∖ 𝐴))) = (vol*‘𝑥)
5		inss1 4184	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
6		simprl 770	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
7	5, 6	sstrid 3941	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∩ 𝐴) ⊆ ℝ)
8		ovolsscl 25414	. . . . . . . . . . . 12 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
9	5, 8	mp3an1 1450	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
10	9	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
11		difss 4083	. . . . . . . . . . 11 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥
12	11, 6	sstrid 3941	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∖ 𝐴) ⊆ ℝ)
13		ovolsscl 25414	. . . . . . . . . . . 12 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
14	11, 13	mp3an1 1450	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
15	14	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
16		ovolun 25427	. . . . . . . . . 10 ⊢ ((((𝑥 ∩ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ) ∧ ((𝑥 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)) → (vol*‘((𝑥 ∩ 𝐴) ∪ (𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
17	7, 10, 12, 15, 16	syl22anc 838	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∩ 𝐴) ∪ (𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
18	4, 17	eqbrtrrid 5125	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
19		simprr 772	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
20	10, 15	readdcld 11141	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ)
21	19, 20	letri3d 11255	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ↔ ((vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ∧ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
22	18, 21	mpbirand 707	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ↔ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
23	22	expr 456	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ↔ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
24	23	pm5.74d 273	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ⊆ ℝ) → (((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
25	2, 24	sylan2 593	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → (((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
26	25	ralbidva 3153	. . 3 ⊢ (𝐴 ⊆ ℝ → (∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
27	26	pm5.32i 574	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
28	1, 27	bitri 275	1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547   class class class wbr 5089  dom cdm 5614  ‘cfv 6481  (class class class)co 7346  ℝcr 11005   + caddc 11009   ≤ cle 11147  vol*covol 25390  volcvol 25391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-q 12847  df-rp 12891  df-ioo 13249  df-ico 13251  df-icc 13252  df-fz 13408  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 25392  df-vol 25393

This theorem is referenced by:  nulmbl  25463  nulmbl2  25464  unmbl  25465  ioombl1  25490  uniioombl  25517  ismblfin  37709  ismbl3  46032