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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > carageneld | Structured version Visualization version GIF version |
Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
carageneld.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
carageneld.x | ⊢ 𝑋 = ∪ dom 𝑂 |
carageneld.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
carageneld.e | ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) |
carageneld.a | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
Ref | Expression |
---|---|
carageneld | ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carageneld.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) | |
2 | carageneld.x | . . . . 5 ⊢ 𝑋 = ∪ dom 𝑂 | |
3 | 2 | pweqi 4515 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
4 | 1, 3 | eleqtrdi 2900 | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂) |
5 | simpl 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑) | |
6 | 3 | eleq2i 2881 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ∈ 𝒫 ∪ dom 𝑂) |
7 | 6 | bicomi 227 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝑎 ∈ 𝒫 𝑋) |
8 | 7 | biimpi 219 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ∈ 𝒫 𝑋) |
9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ∈ 𝒫 𝑋) |
10 | carageneld.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) | |
11 | 5, 9, 10 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
12 | 11 | ralrimiva 3149 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
13 | 4, 12 | jca 515 | . 2 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))) |
14 | carageneld.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
15 | carageneld.s | . . 3 ⊢ 𝑆 = (CaraGen‘𝑂) | |
16 | 14, 15 | caragenel 43134 | . 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))) |
17 | 13, 16 | mpbird 260 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∖ cdif 3878 ∩ cin 3880 𝒫 cpw 4497 ∪ cuni 4800 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 +𝑒 cxad 12493 OutMeascome 43128 CaraGenccaragen 43130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-caragen 43131 |
This theorem is referenced by: caragen0 43145 caragenunidm 43147 caragenuncl 43152 caragendifcl 43153 carageniuncl 43162 caragenel2d 43171 |
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