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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 4112 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
4 | 1, 3 | syl6eleq 2698 | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂) |
5 | simpl 472 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑) | |
6 | 3 | eleq2i 2680 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ∈ 𝒫 ∪ dom 𝑂) |
7 | 6 | bicomi 213 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝑎 ∈ 𝒫 𝑋) |
8 | 7 | biimpi 205 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ∈ 𝒫 𝑋) |
9 | 8 | adantl 481 | . . . . 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 691 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
12 | 11 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
13 | 4, 12 | jca 553 | . 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 39385 | . 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 246 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∖ cdif 3537 ∩ cin 3539 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: caragen0 39396 caragenunidm 39398 caragenuncl 39403 caragendifcl 39404 carageniuncl 39413 caragenel2d 39422 |
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