Theorem ismbl2 25507

Description: From ovolun 25479, 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 25506	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))
2		elpwi 4549	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3		inundif 4420	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ∪ (𝑥 ∖ 𝐴)) = 𝑥
4	3	fveq2i 6838	. . . . . . . . 9 ⊢ (vol*‘((𝑥 ∩ 𝐴) ∪ (𝑥 ∖ 𝐴))) = (vol*‘𝑥)
5		inss1 4178	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
6		simprl 771	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
7	5, 6	sstrid 3934	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∩ 𝐴) ⊆ ℝ)
8		ovolsscl 25466	. . . . . . . . . . . 12 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
9	5, 8	mp3an1 1451	. . . . . . . . . . 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 4077	. . . . . . . . . . 11 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥
12	11, 6	sstrid 3934	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∖ 𝐴) ⊆ ℝ)
13		ovolsscl 25466	. . . . . . . . . . . 12 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
14	11, 13	mp3an1 1451	. . . . . . . . . . 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 25479	. . . . . . . . . 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 839	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∩ 𝐴) ∪ (𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
18	4, 17	eqbrtrrid 5122	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
19		simprr 773	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
20	10, 15	readdcld 11168	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ)
21	19, 20	letri3d 11282	. . . . . . . 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 708	. . . . . . 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 594	. . . 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 3159	. . 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 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542   class class class wbr 5086  dom cdm 5625  ‘cfv 6493  (class class class)co 7361  ℝcr 11031   + caddc 11035   ≤ cle 11174  vol*covol 25442  volcvol 25443

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 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-n0 12432  df-z 12519  df-uz 12783  df-q 12893  df-rp 12937  df-ioo 13296  df-ico 13298  df-icc 13299  df-fz 13456  df-fl 13745  df-seq 13958  df-exp 14018  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-ovol 25444  df-vol 25445

This theorem is referenced by:  nulmbl  25515  nulmbl2  25516  unmbl  25517  ioombl1  25542  uniioombl  25569  ismblfin  37999  ismbl3  46435