![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 33767 | . . 3 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) |
4 | 3 | eleq2d 2818 | . 2 ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})) |
5 | ineq2 4206 | . . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑒 ∩ 𝑎) = (𝑒 ∩ 𝐴)) | |
6 | 5 | fveq2d 6895 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝑒 ∩ 𝑎)) = (𝑀‘(𝑒 ∩ 𝐴))) |
7 | difeq2 4116 | . . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑒 ∖ 𝑎) = (𝑒 ∖ 𝐴)) | |
8 | 7 | fveq2d 6895 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝑒 ∖ 𝑎)) = (𝑀‘(𝑒 ∖ 𝐴))) |
9 | 6, 8 | oveq12d 7430 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = ((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴)))) |
10 | 9 | eqeq1d 2733 | . . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒) ↔ ((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒))) |
11 | 10 | ralbidv 3176 | . . . 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 3683 | . . 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 3492 | . . . . . 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 4605 | . . . . . 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 629 | . . 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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 Vcvv 3473 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 0cc0 11116 +∞cpnf 11252 +𝑒 cxad 13097 [,]cicc 13334 toCaraSigaccarsg 33765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-carsg 33766 |
This theorem is referenced by: baselcarsg 33770 0elcarsg 33771 difelcarsg 33774 inelcarsg 33775 carsgclctunlem1 33781 carsgclctunlem2 33783 carsgclctun 33785 |
Copyright terms: Public domain | W3C validator |