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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0elcarsg | Structured version Visualization version GIF version |
Description: The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
baselcarsg.1 | ⊢ (𝜑 → (𝑀‘∅) = 0) |
Ref | Expression |
---|---|
0elcarsg | ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4304 | . . 3 ⊢ ∅ ⊆ 𝑂 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝑂) |
3 | in0 4299 | . . . . . . . 8 ⊢ (𝑒 ∩ ∅) = ∅ | |
4 | 3 | fveq2i 6648 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∩ ∅)) = (𝑀‘∅) |
5 | baselcarsg.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
6 | 4, 5 | syl5eq 2845 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∩ ∅)) = 0) |
7 | dif0 4286 | . . . . . . . 8 ⊢ (𝑒 ∖ ∅) = 𝑒 | |
8 | 7 | fveq2i 6648 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒) |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒)) |
10 | 6, 9 | oveq12d 7153 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
11 | 10 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
12 | iccssxr 12808 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
13 | carsgval.2 | . . . . . . 7 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
14 | 13 | ffvelrnda 6828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
15 | 12, 14 | sseldi 3913 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
16 | xaddid2 12623 | . . . . 5 ⊢ ((𝑀‘𝑒) ∈ ℝ* → (0 +𝑒 (𝑀‘𝑒)) = (𝑀‘𝑒)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (0 +𝑒 (𝑀‘𝑒)) = (𝑀‘𝑒)) |
18 | 11, 17 | eqtrd 2833 | . . 3 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
19 | 18 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
20 | carsgval.1 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
21 | 20, 13 | elcarsg 31673 | . 2 ⊢ (𝜑 → (∅ ∈ (toCaraSiga‘𝑀) ↔ (∅ ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)))) |
22 | 2, 19, 21 | mpbir2and 712 | 1 ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 +𝑒 cxad 12493 [,]cicc 12729 toCaraSigaccarsg 31669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-xadd 12496 df-icc 12733 df-carsg 31670 |
This theorem is referenced by: carsggect 31686 omsmeas 31691 |
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