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Mirrors > Home > MPE Home > Th. List > Mathboxes > baselcarsg | Structured version Visualization version GIF version |
Description: The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
baselcarsg.1 | ⊢ (𝜑 → (𝑀‘∅) = 0) |
Ref | Expression |
---|---|
baselcarsg | ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 4018 | . . 3 ⊢ (𝜑 → 𝑂 ⊆ 𝑂) | |
2 | elpwi 4611 | . . . . . . . . 9 ⊢ (𝑒 ∈ 𝒫 𝑂 → 𝑒 ⊆ 𝑂) | |
3 | 2 | adantl 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑒 ⊆ 𝑂) |
4 | dfss2 3980 | . . . . . . . 8 ⊢ (𝑒 ⊆ 𝑂 ↔ (𝑒 ∩ 𝑂) = 𝑒) | |
5 | 3, 4 | sylib 218 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑒 ∩ 𝑂) = 𝑒) |
6 | 5 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∩ 𝑂)) = (𝑀‘𝑒)) |
7 | ssdif0 4371 | . . . . . . . . 9 ⊢ (𝑒 ⊆ 𝑂 ↔ (𝑒 ∖ 𝑂) = ∅) | |
8 | 3, 7 | sylib 218 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑒 ∖ 𝑂) = ∅) |
9 | 8 | fveq2d 6910 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ 𝑂)) = (𝑀‘∅)) |
10 | baselcarsg.1 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
11 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘∅) = 0) |
12 | 9, 11 | eqtrd 2774 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ 𝑂)) = 0) |
13 | 6, 12 | oveq12d 7448 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = ((𝑀‘𝑒) +𝑒 0)) |
14 | iccssxr 13466 | . . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | carsgval.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
16 | 15 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
17 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑒 ∈ 𝒫 𝑂) | |
18 | 16, 17 | ffvelcdmd 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
19 | 14, 18 | sselid 3992 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
20 | xaddrid 13279 | . . . . . 6 ⊢ ((𝑀‘𝑒) ∈ ℝ* → ((𝑀‘𝑒) +𝑒 0) = (𝑀‘𝑒)) | |
21 | 19, 20 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘𝑒) +𝑒 0) = (𝑀‘𝑒)) |
22 | 13, 21 | eqtrd 2774 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)) |
23 | 22 | ralrimiva 3143 | . . 3 ⊢ (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)) |
24 | 1, 23 | jca 511 | . 2 ⊢ (𝜑 → (𝑂 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒))) |
25 | carsgval.1 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
26 | 25, 15 | elcarsg 34286 | . 2 ⊢ (𝜑 → (𝑂 ∈ (toCaraSiga‘𝑀) ↔ (𝑂 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)))) |
27 | 24, 26 | mpbird 257 | 1 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∖ cdif 3959 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 0cc0 11152 +∞cpnf 11289 ℝ*cxr 11291 +𝑒 cxad 13149 [,]cicc 13386 toCaraSigaccarsg 34282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-xadd 13152 df-icc 13390 df-carsg 34283 |
This theorem is referenced by: carsguni 34289 fiunelcarsg 34297 carsgsiga 34303 |
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