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 4349 | . . 3 ⊢ ∅ ⊆ 𝑂 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝑂) |
3 | in0 4344 | . . . . . . . 8 ⊢ (𝑒 ∩ ∅) = ∅ | |
4 | 3 | fveq2i 6667 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∩ ∅)) = (𝑀‘∅) |
5 | baselcarsg.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
6 | 4, 5 | syl5eq 2868 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∩ ∅)) = 0) |
7 | dif0 4331 | . . . . . . . 8 ⊢ (𝑒 ∖ ∅) = 𝑒 | |
8 | 7 | fveq2i 6667 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒) |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒)) |
10 | 6, 9 | oveq12d 7168 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
11 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
12 | iccssxr 12813 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
13 | carsgval.2 | . . . . . . 7 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
14 | 13 | ffvelrnda 6845 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
15 | 12, 14 | sseldi 3964 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
16 | xaddid2 12629 | . . . . 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 2856 | . . 3 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
19 | 18 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
20 | carsgval.1 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
21 | 20, 13 | elcarsg 31558 | . 2 ⊢ (𝜑 → (∅ ∈ (toCaraSiga‘𝑀) ↔ (∅ ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)))) |
22 | 2, 19, 21 | mpbir2and 711 | 1 ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 +𝑒 cxad 12499 [,]cicc 12735 toCaraSigaccarsg 31554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-xadd 12502 df-icc 12739 df-carsg 31555 |
This theorem is referenced by: carsggect 31571 omsmeas 31576 |
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