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Mirrors > Home > MPE Home > Th. List > mblsplit | Structured version Visualization version GIF version |
Description: The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.) |
Ref | Expression |
---|---|
mblsplit | ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reex 11150 | . . . 4 ⊢ ℝ ∈ V | |
2 | 1 | elpw2 5306 | . . 3 ⊢ (𝐵 ∈ 𝒫 ℝ ↔ 𝐵 ⊆ ℝ) |
3 | ismbl 24913 | . . . 4 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) | |
4 | fveq2 6846 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (vol*‘𝑥) = (vol*‘𝐵)) | |
5 | 4 | eleq1d 2819 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ)) |
6 | ineq1 4169 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
7 | 6 | fveq2d 6850 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (vol*‘(𝑥 ∩ 𝐴)) = (vol*‘(𝐵 ∩ 𝐴))) |
8 | difeq1 4079 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∖ 𝐴) = (𝐵 ∖ 𝐴)) | |
9 | 8 | fveq2d 6850 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (vol*‘(𝑥 ∖ 𝐴)) = (vol*‘(𝐵 ∖ 𝐴))) |
10 | 7, 9 | oveq12d 7379 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴)))) |
11 | 4, 10 | eqeq12d 2749 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ↔ (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴))))) |
12 | 5, 11 | imbi12d 345 | . . . . 5 ⊢ (𝑥 = 𝐵 → (((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ ((vol*‘𝐵) ∈ ℝ → (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴)))))) |
13 | 12 | rspccv 3580 | . . . 4 ⊢ (∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) → (𝐵 ∈ 𝒫 ℝ → ((vol*‘𝐵) ∈ ℝ → (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴)))))) |
14 | 3, 13 | simplbiim 506 | . . 3 ⊢ (𝐴 ∈ dom vol → (𝐵 ∈ 𝒫 ℝ → ((vol*‘𝐵) ∈ ℝ → (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴)))))) |
15 | 2, 14 | biimtrrid 242 | . 2 ⊢ (𝐴 ∈ dom vol → (𝐵 ⊆ ℝ → ((vol*‘𝐵) ∈ ℝ → (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴)))))) |
16 | 15 | 3imp 1112 | 1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) = ((vol*‘(𝐵 ∩ 𝐴)) + (vol*‘(𝐵 ∖ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∖ cdif 3911 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4564 dom cdm 5637 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 + caddc 11062 vol*covol 24849 volcvol 24850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-ico 13279 df-icc 13280 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-ovol 24851 df-vol 24852 |
This theorem is referenced by: cmmbl 24921 nulmbl2 24923 unmbl 24924 shftmbl 24925 volun 24932 voliunlem1 24937 uniioombllem4 24973 uniioombllem5 24974 mblfinlem3 36167 mblfinlem4 36168 |
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