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 30371 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑂) |
5 | dfss4 3858 | . . . 4 ⊢ (𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ 𝐴)) = 𝐴) | |
6 | 4, 5 | sylib 208 | . . 3 ⊢ (𝜑 → (𝑂 ∖ (𝑂 ∖ 𝐴)) = 𝐴) |
7 | inelcarsg.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) | |
8 | 1, 2, 7 | elcarsgss 30371 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑂) |
9 | dfss4 3858 | . . . 4 ⊢ (𝐵 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ 𝐵)) = 𝐵) | |
10 | 8, 9 | sylib 208 | . . 3 ⊢ (𝜑 → (𝑂 ∖ (𝑂 ∖ 𝐵)) = 𝐵) |
11 | 6, 10 | uneq12d 3768 | . 2 ⊢ (𝜑 → ((𝑂 ∖ (𝑂 ∖ 𝐴)) ∪ (𝑂 ∖ (𝑂 ∖ 𝐵))) = (𝐴 ∪ 𝐵)) |
12 | difindi 3881 | . . 3 ⊢ (𝑂 ∖ ((𝑂 ∖ 𝐴) ∩ (𝑂 ∖ 𝐵))) = ((𝑂 ∖ (𝑂 ∖ 𝐴)) ∪ (𝑂 ∖ (𝑂 ∖ 𝐵))) | |
13 | 1, 2, 3 | difelcarsg 30372 | . . . . 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 30372 | . . . . 5 ⊢ (𝜑 → (𝑂 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) |
16 | 1, 2, 13, 14, 15 | inelcarsg 30373 | . . . 4 ⊢ (𝜑 → ((𝑂 ∖ 𝐴) ∩ (𝑂 ∖ 𝐵)) ∈ (toCaraSiga‘𝑀)) |
17 | 1, 2, 16 | difelcarsg 30372 | . . 3 ⊢ (𝜑 → (𝑂 ∖ ((𝑂 ∖ 𝐴) ∩ (𝑂 ∖ 𝐵))) ∈ (toCaraSiga‘𝑀)) |
18 | 12, 17 | syl5eqelr 2706 | . 2 ⊢ (𝜑 → ((𝑂 ∖ (𝑂 ∖ 𝐴)) ∪ (𝑂 ∖ (𝑂 ∖ 𝐵))) ∈ (toCaraSiga‘𝑀)) |
19 | 11, 18 | eqeltrrd 2702 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 ≤ cle 10075 +𝑒 cxad 11944 [,]cicc 12178 toCaraSigaccarsg 30363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-xadd 11947 df-icc 12182 df-carsg 30364 |
This theorem is referenced by: fiunelcarsg 30378 |
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