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| Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenel | Structured version Visualization version GIF version | ||
| Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| caragenel.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
| caragenel.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
| Ref | Expression |
|---|---|
| caragenel | ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | caragenel.s | . . . 4 ⊢ 𝑆 = (CaraGen‘𝑂) | |
| 2 | caragenel.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
| 3 | caragenval 46508 | . . . . 5 ⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |
| 5 | 1, 4 | eqtrid 2789 | . . 3 ⊢ (𝜑 → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |
| 6 | 5 | eleq2d 2827 | . 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ 𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})) |
| 7 | ineq2 4214 | . . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∩ 𝑒) = (𝑎 ∩ 𝐸)) | |
| 8 | 7 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝐸))) |
| 9 | difeq2 4120 | . . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∖ 𝑒) = (𝑎 ∖ 𝐸)) | |
| 10 | 9 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝐸))) |
| 11 | 8, 10 | oveq12d 7449 | . . . . . 6 ⊢ (𝑒 = 𝐸 → ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸)))) |
| 12 | 11 | eqeq1d 2739 | . . . . 5 ⊢ (𝑒 = 𝐸 → (((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))) |
| 13 | 12 | ralbidv 3178 | . . . 4 ⊢ (𝑒 = 𝐸 → (∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎) ↔ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))) |
| 14 | 13 | elrab 3692 | . . 3 ⊢ (𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))) |
| 15 | 14 | a1i 11 | . 2 ⊢ (𝜑 → (𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))) |
| 16 | 6, 15 | bitrd 279 | 1 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ∩ cin 3950 𝒫 cpw 4600 ∪ cuni 4907 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 +𝑒 cxad 13152 OutMeascome 46504 CaraGenccaragen 46506 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-caragen 46507 |
| This theorem is referenced by: caragensplit 46515 caragenelss 46516 carageneld 46517 caragendifcl 46529 isvonmbl 46653 |
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