Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragensplit | Structured version Visualization version GIF version |
Description: If 𝐸 is in the set generated by the Caratheodory's method, then it splits any set 𝐴 in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set 𝐴. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragensplit.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragensplit.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
caragensplit.x | ⊢ 𝑋 = ∪ dom 𝑂 |
caragensplit.e | ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
caragensplit.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Ref | Expression |
---|---|
caragensplit | ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragensplit.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
2 | caragensplit.o | . . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
3 | caragensplit.x | . . . . . . 7 ⊢ 𝑋 = ∪ dom 𝑂 | |
4 | 2, 3 | unidmex 41189 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | ssexg 5218 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
6 | 1, 4, 5 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | elpwg 4541 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
9 | 1, 8 | mpbird 258 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
10 | 3 | pweqi 4539 | . . 3 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
11 | 9, 10 | eleqtrdi 2920 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂) |
12 | caragensplit.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | |
13 | caragensplit.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
14 | 2, 13 | caragenel 42654 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))) |
15 | 12, 14 | mpbid 233 | . . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))) |
16 | 15 | simprd 496 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
17 | ineq1 4178 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∩ 𝐸) = (𝐴 ∩ 𝐸)) | |
18 | 17 | fveq2d 6667 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))) |
19 | difeq1 4089 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∖ 𝐸) = (𝐴 ∖ 𝐸)) | |
20 | 19 | fveq2d 6667 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∖ 𝐸)) = (𝑂‘(𝐴 ∖ 𝐸))) |
21 | 18, 20 | oveq12d 7163 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
22 | fveq2 6663 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑂‘𝑎) = (𝑂‘𝐴)) | |
23 | 21, 22 | eqeq12d 2834 | . . 3 ⊢ (𝑎 = 𝐴 → (((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))) |
24 | 23 | rspcva 3618 | . 2 ⊢ ((𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) |
25 | 11, 16, 24 | syl2anc 584 | 1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 +𝑒 cxad 12493 OutMeascome 42648 CaraGenccaragen 42650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-caragen 42651 |
This theorem is referenced by: caragenuncllem 42671 carageniuncllem1 42680 carageniuncllem2 42681 caratheodorylem1 42685 |
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