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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elcarsg | Structured version Visualization version GIF version | ||
| Description: Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.) |
| Ref | Expression |
|---|---|
| carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
| carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
| Ref | Expression |
|---|---|
| elcarsg | ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | carsgval.1 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
| 2 | carsgval.2 | . . . 4 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
| 3 | 1, 2 | carsgval 34305 | . . 3 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) |
| 4 | 3 | eleq2d 2827 | . 2 ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})) |
| 5 | ineq2 4214 | . . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑒 ∩ 𝑎) = (𝑒 ∩ 𝐴)) | |
| 6 | 5 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝑒 ∩ 𝑎)) = (𝑀‘(𝑒 ∩ 𝐴))) |
| 7 | difeq2 4120 | . . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑒 ∖ 𝑎) = (𝑒 ∖ 𝐴)) | |
| 8 | 7 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝑒 ∖ 𝑎)) = (𝑀‘(𝑒 ∖ 𝐴))) |
| 9 | 6, 8 | oveq12d 7449 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = ((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴)))) |
| 10 | 9 | eqeq1d 2739 | . . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒) ↔ ((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒))) |
| 11 | 10 | ralbidv 3178 | . . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒) ↔ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒))) |
| 12 | 11 | elrab 3692 | . . 3 ⊢ (𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ↔ (𝐴 ∈ 𝒫 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒))) |
| 13 | elex 3501 | . . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝑂 → 𝐴 ∈ V) | |
| 14 | 13 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑂 → 𝐴 ∈ V)) |
| 15 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝑂) → 𝑂 ∈ 𝑉) |
| 16 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝑂) → 𝐴 ⊆ 𝑂) | |
| 17 | 15, 16 | ssexd 5324 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝑂) → 𝐴 ∈ V) |
| 18 | 17 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝐴 ⊆ 𝑂 → 𝐴 ∈ V)) |
| 19 | elpwg 4603 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 ⊆ 𝑂)) | |
| 20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 ⊆ 𝑂))) |
| 21 | 14, 18, 20 | pm5.21ndd 379 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 ⊆ 𝑂)) |
| 22 | 21 | anbi1d 631 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝒫 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)))) |
| 23 | 12, 22 | bitrid 283 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)))) |
| 24 | 4, 23 | bitrd 279 | 1 ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 +𝑒 cxad 13152 [,]cicc 13390 toCaraSigaccarsg 34303 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-iun 4993 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-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-carsg 34304 |
| This theorem is referenced by: baselcarsg 34308 0elcarsg 34309 difelcarsg 34312 inelcarsg 34313 carsgclctunlem1 34319 carsgclctunlem2 34321 carsgclctun 34323 |
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