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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 4560 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
4 | 1, 3 | eleqtrdi 2926 | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂) |
5 | simpl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑) | |
6 | 3 | eleq2i 2907 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ∈ 𝒫 ∪ dom 𝑂) |
7 | 6 | bicomi 226 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝑎 ∈ 𝒫 𝑋) |
8 | 7 | biimpi 218 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ∈ 𝒫 𝑋) |
9 | 8 | adantl 484 | . . . . 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 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
12 | 11 | ralrimiva 3185 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
13 | 4, 12 | jca 514 | . 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 42784 | . 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 259 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∖ cdif 3936 ∩ cin 3938 𝒫 cpw 4542 ∪ cuni 4841 dom cdm 5558 ‘cfv 6358 (class class class)co 7159 +𝑒 cxad 12508 OutMeascome 42778 CaraGenccaragen 42780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-caragen 42781 |
This theorem is referenced by: caragen0 42795 caragenunidm 42797 caragenuncl 42802 caragendifcl 42803 carageniuncl 42812 caragenel2d 42821 |
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