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 38242 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | ssexg 4732 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
6 | 1, 4, 5 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | elpwg 4116 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
9 | 1, 8 | mpbird 246 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
10 | 3 | pweqi 4112 | . . 3 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
11 | 9, 10 | syl6eleq 2698 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂) |
12 | caragensplit.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | |
13 | caragensplit.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
14 | 2, 13 | caragenel 39385 | . . . 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 221 | . . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))) |
16 | 15 | simprd 478 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
17 | ineq1 3769 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∩ 𝐸) = (𝐴 ∩ 𝐸)) | |
18 | 17 | fveq2d 6107 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))) |
19 | difeq1 3683 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∖ 𝐸) = (𝐴 ∖ 𝐸)) | |
20 | 19 | fveq2d 6107 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∖ 𝐸)) = (𝑂‘(𝐴 ∖ 𝐸))) |
21 | 18, 20 | oveq12d 6567 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
22 | fveq2 6103 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑂‘𝑎) = (𝑂‘𝐴)) | |
23 | 21, 22 | eqeq12d 2625 | . . 3 ⊢ (𝑎 = 𝐴 → (((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))) |
24 | 23 | rspcva 3280 | . 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 691 | 1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 +𝑒 cxad 11820 OutMeascome 39379 CaraGenccaragen 39381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-caragen 39382 |
This theorem is referenced by: caragenuncllem 39402 carageniuncllem1 39411 carageniuncllem2 39412 caratheodorylem1 39416 |
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