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Theorem ismbl2 25576
Description: From ovolun 25548, it suffices to show that the measure of 𝑥 is at least the sum of the measures of 𝑥𝐴 and 𝑥𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
Assertion
Ref Expression
ismbl2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl2
StepHypRef Expression
1 ismbl 25575 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
2 elpwi 4612 . . . . 5 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3 inundif 4485 . . . . . . . . . 10 ((𝑥𝐴) ∪ (𝑥𝐴)) = 𝑥
43fveq2i 6910 . . . . . . . . 9 (vol*‘((𝑥𝐴) ∪ (𝑥𝐴))) = (vol*‘𝑥)
5 inss1 4245 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
6 simprl 771 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
75, 6sstrid 4007 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
8 ovolsscl 25535 . . . . . . . . . . . 12 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
95, 8mp3an1 1447 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
109adantl 481 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
11 difss 4146 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
1211, 6sstrid 4007 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
13 ovolsscl 25535 . . . . . . . . . . . 12 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1411, 13mp3an1 1447 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1514adantl 481 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
16 ovolun 25548 . . . . . . . . . 10 ((((𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) ∧ ((𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ (𝑥𝐴))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
177, 10, 12, 15, 16syl22anc 839 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ (𝑥𝐴))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
184, 17eqbrtrrid 5184 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
19 simprr 773 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
2010, 15readdcld 11288 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ∈ ℝ)
2119, 20letri3d 11401 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ↔ ((vol*‘𝑥) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ∧ ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
2218, 21mpbirand 707 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ↔ ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
2322expr 456 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ↔ ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
2423pm5.74d 273 . . . . 5 ((𝐴 ⊆ ℝ ∧ 𝑥 ⊆ ℝ) → (((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
252, 24sylan2 593 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → (((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
2625ralbidva 3174 . . 3 (𝐴 ⊆ ℝ → (∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
2726pm5.32i 574 . 2 ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
281, 27bitri 275 1 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cdif 3960  cun 3961  cin 3962  wss 3963  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689  cfv 6563  (class class class)co 7431  cr 11152   + caddc 11156  cle 11294  vol*covol 25511  volcvol 25512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fl 13829  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-ovol 25513  df-vol 25514
This theorem is referenced by:  nulmbl  25584  nulmbl2  25585  unmbl  25586  ioombl1  25611  uniioombl  25638  ismblfin  37648  ismbl3  45942
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