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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 3914 | . . 3 ⊢ (𝜑 → 𝑂 ⊆ 𝑂) | |
2 | elpwi 4512 | . . . . . . . . 9 ⊢ (𝑒 ∈ 𝒫 𝑂 → 𝑒 ⊆ 𝑂) | |
3 | 2 | adantl 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑒 ⊆ 𝑂) |
4 | df-ss 3874 | . . . . . . . 8 ⊢ (𝑒 ⊆ 𝑂 ↔ (𝑒 ∩ 𝑂) = 𝑒) | |
5 | 3, 4 | sylib 221 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑒 ∩ 𝑂) = 𝑒) |
6 | 5 | fveq2d 6710 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∩ 𝑂)) = (𝑀‘𝑒)) |
7 | ssdif0 4268 | . . . . . . . . 9 ⊢ (𝑒 ⊆ 𝑂 ↔ (𝑒 ∖ 𝑂) = ∅) | |
8 | 3, 7 | sylib 221 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑒 ∖ 𝑂) = ∅) |
9 | 8 | fveq2d 6710 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ 𝑂)) = (𝑀‘∅)) |
10 | baselcarsg.1 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
11 | 10 | adantr 484 | . . . . . . 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 7220 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = ((𝑀‘𝑒) +𝑒 0)) |
14 | iccssxr 13001 | . . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | carsgval.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
16 | 15 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
17 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑒 ∈ 𝒫 𝑂) | |
18 | 16, 17 | ffvelrnd 6894 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
19 | 14, 18 | sseldi 3889 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
20 | xaddid1 12814 | . . . . . 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 3098 | . . 3 ⊢ (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)) |
24 | 1, 23 | jca 515 | . 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 31956 | . 2 ⊢ (𝜑 → (𝑂 ∈ (toCaraSiga‘𝑀) ↔ (𝑂 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)))) |
27 | 24, 26 | mpbird 260 | 1 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ∖ cdif 3854 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 0cc0 10712 +∞cpnf 10847 ℝ*cxr 10849 +𝑒 cxad 12685 [,]cicc 12921 toCaraSigaccarsg 31952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-xadd 12688 df-icc 12925 df-carsg 31953 |
This theorem is referenced by: carsguni 31959 fiunelcarsg 31967 carsgsiga 31973 |
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