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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 4005 | . . 3 ⊢ ∅ ⊆ 𝑂 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝑂) |
3 | in0 4001 | . . . . . . . 8 ⊢ (𝑒 ∩ ∅) = ∅ | |
4 | 3 | fveq2i 6232 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∩ ∅)) = (𝑀‘∅) |
5 | baselcarsg.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
6 | 4, 5 | syl5eq 2697 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∩ ∅)) = 0) |
7 | dif0 3983 | . . . . . . . 8 ⊢ (𝑒 ∖ ∅) = 𝑒 | |
8 | 7 | fveq2i 6232 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒) |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒)) |
10 | 6, 9 | oveq12d 6708 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
12 | iccssxr 12294 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
13 | carsgval.2 | . . . . . . 7 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
14 | 13 | ffvelrnda 6399 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
15 | 12, 14 | sseldi 3634 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
16 | xaddid2 12111 | . . . . 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 2685 | . . 3 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
19 | 18 | ralrimiva 2995 | . 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
20 | carsgval.1 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
21 | 20, 13 | elcarsg 30495 | . 2 ⊢ (𝜑 → (∅ ∈ (toCaraSiga‘𝑀) ↔ (∅ ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)))) |
22 | 2, 19, 21 | mpbir2and 977 | 1 ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∖ cdif 3604 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 𝒫 cpw 4191 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 0cc0 9974 +∞cpnf 10109 ℝ*cxr 10111 +𝑒 cxad 11982 [,]cicc 12216 toCaraSigaccarsg 30491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-xadd 11985 df-icc 12220 df-carsg 30492 |
This theorem is referenced by: carsggect 30508 omsmeas 30513 |
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