Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 6672 | . . . . . . 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 6672 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒) |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒)) |
10 | 6, 9 | oveq12d 7173 | . . . . 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 12818 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
13 | carsgval.2 | . . . . . . 7 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
14 | 13 | ffvelrnda 6850 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
15 | 12, 14 | sseldi 3964 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
16 | xaddid2 12634 | . . . . 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 31563 | . 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 6350 ‘cfv 6354 (class class class)co 7155 0cc0 10536 +∞cpnf 10671 ℝ*cxr 10673 +𝑒 cxad 12504 [,]cicc 12740 toCaraSigaccarsg 31559 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-xadd 12507 df-icc 12744 df-carsg 31560 |
This theorem is referenced by: carsggect 31576 omsmeas 31581 |
Copyright terms: Public domain | W3C validator |