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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 39708 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | ssexg 4948 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
6 | 1, 4, 5 | syl2anc 696 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | elpwg 4302 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
9 | 1, 8 | mpbird 247 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
10 | 3 | pweqi 4298 | . . 3 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
11 | 9, 10 | syl6eleq 2841 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂) |
12 | caragensplit.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | |
13 | caragensplit.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
14 | 2, 13 | caragenel 41207 | . . . 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 222 | . . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))) |
16 | 15 | simprd 482 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
17 | ineq1 3942 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∩ 𝐸) = (𝐴 ∩ 𝐸)) | |
18 | 17 | fveq2d 6348 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))) |
19 | difeq1 3856 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∖ 𝐸) = (𝐴 ∖ 𝐸)) | |
20 | 19 | fveq2d 6348 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∖ 𝐸)) = (𝑂‘(𝐴 ∖ 𝐸))) |
21 | 18, 20 | oveq12d 6823 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
22 | fveq2 6344 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑂‘𝑎) = (𝑂‘𝐴)) | |
23 | 21, 22 | eqeq12d 2767 | . . 3 ⊢ (𝑎 = 𝐴 → (((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))) |
24 | 23 | rspcva 3439 | . 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 696 | 1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1624 ∈ wcel 2131 ∀wral 3042 Vcvv 3332 ∖ cdif 3704 ∩ cin 3706 ⊆ wss 3707 𝒫 cpw 4294 ∪ cuni 4580 dom cdm 5258 ‘cfv 6041 (class class class)co 6805 +𝑒 cxad 12129 OutMeascome 41201 CaraGenccaragen 41203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ral 3047 df-rex 3048 df-rab 3051 df-v 3334 df-sbc 3569 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-iota 6004 df-fun 6043 df-fv 6049 df-ov 6808 df-caragen 41204 |
This theorem is referenced by: caragenuncllem 41224 carageniuncllem1 41233 carageniuncllem2 41234 caratheodorylem1 41238 |
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