Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unelcarsg | Structured version Visualization version GIF version |
Description: The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
difelcarsg.1 | ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) |
inelcarsg.1 | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) |
inelcarsg.2 | ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) |
Ref | Expression |
---|---|
unelcarsg | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carsgval.1 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
2 | carsgval.2 | . . . . 5 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
3 | difelcarsg.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) | |
4 | 1, 2, 3 | elcarsgss 32272 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑂) |
5 | dfss4 4198 | . . . 4 ⊢ (𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ 𝐴)) = 𝐴) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ (𝜑 → (𝑂 ∖ (𝑂 ∖ 𝐴)) = 𝐴) |
7 | inelcarsg.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) | |
8 | 1, 2, 7 | elcarsgss 32272 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑂) |
9 | dfss4 4198 | . . . 4 ⊢ (𝐵 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ 𝐵)) = 𝐵) | |
10 | 8, 9 | sylib 217 | . . 3 ⊢ (𝜑 → (𝑂 ∖ (𝑂 ∖ 𝐵)) = 𝐵) |
11 | 6, 10 | uneq12d 4103 | . 2 ⊢ (𝜑 → ((𝑂 ∖ (𝑂 ∖ 𝐴)) ∪ (𝑂 ∖ (𝑂 ∖ 𝐵))) = (𝐴 ∪ 𝐵)) |
12 | difindi 4221 | . . 3 ⊢ (𝑂 ∖ ((𝑂 ∖ 𝐴) ∩ (𝑂 ∖ 𝐵))) = ((𝑂 ∖ (𝑂 ∖ 𝐴)) ∪ (𝑂 ∖ (𝑂 ∖ 𝐵))) | |
13 | 1, 2, 3 | difelcarsg 32273 | . . . . 5 ⊢ (𝜑 → (𝑂 ∖ 𝐴) ∈ (toCaraSiga‘𝑀)) |
14 | inelcarsg.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) | |
15 | 1, 2, 7 | difelcarsg 32273 | . . . . 5 ⊢ (𝜑 → (𝑂 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) |
16 | 1, 2, 13, 14, 15 | inelcarsg 32274 | . . . 4 ⊢ (𝜑 → ((𝑂 ∖ 𝐴) ∩ (𝑂 ∖ 𝐵)) ∈ (toCaraSiga‘𝑀)) |
17 | 1, 2, 16 | difelcarsg 32273 | . . 3 ⊢ (𝜑 → (𝑂 ∖ ((𝑂 ∖ 𝐴) ∩ (𝑂 ∖ 𝐵))) ∈ (toCaraSiga‘𝑀)) |
18 | 12, 17 | eqeltrrid 2846 | . 2 ⊢ (𝜑 → ((𝑂 ∖ (𝑂 ∖ 𝐴)) ∪ (𝑂 ∖ (𝑂 ∖ 𝐵))) ∈ (toCaraSiga‘𝑀)) |
19 | 11, 18 | eqeltrrd 2842 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∖ cdif 3889 ∪ cun 3890 ∩ cin 3891 ⊆ wss 3892 𝒫 cpw 4539 class class class wbr 5079 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 0cc0 10872 +∞cpnf 11007 ≤ cle 11011 +𝑒 cxad 12845 [,]cicc 13081 toCaraSigaccarsg 32264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-xadd 12848 df-icc 13085 df-carsg 32265 |
This theorem is referenced by: fiunelcarsg 32279 |
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