Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 3987 | . . 3 ⊢ (𝜑 → 𝑂 ⊆ 𝑂) | |
2 | elpwi 4547 | . . . . . . . . 9 ⊢ (𝑒 ∈ 𝒫 𝑂 → 𝑒 ⊆ 𝑂) | |
3 | 2 | adantl 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑒 ⊆ 𝑂) |
4 | df-ss 3949 | . . . . . . . 8 ⊢ (𝑒 ⊆ 𝑂 ↔ (𝑒 ∩ 𝑂) = 𝑒) | |
5 | 3, 4 | sylib 219 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑒 ∩ 𝑂) = 𝑒) |
6 | 5 | fveq2d 6667 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∩ 𝑂)) = (𝑀‘𝑒)) |
7 | ssdif0 4320 | . . . . . . . . 9 ⊢ (𝑒 ⊆ 𝑂 ↔ (𝑒 ∖ 𝑂) = ∅) | |
8 | 3, 7 | sylib 219 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑒 ∖ 𝑂) = ∅) |
9 | 8 | fveq2d 6667 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ 𝑂)) = (𝑀‘∅)) |
10 | baselcarsg.1 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
11 | 10 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘∅) = 0) |
12 | 9, 11 | eqtrd 2853 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ 𝑂)) = 0) |
13 | 6, 12 | oveq12d 7163 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = ((𝑀‘𝑒) +𝑒 0)) |
14 | iccssxr 12807 | . . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | carsgval.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
16 | 15 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
17 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → 𝑒 ∈ 𝒫 𝑂) | |
18 | 16, 17 | ffvelrnd 6844 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
19 | 14, 18 | sseldi 3962 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
20 | xaddid1 12622 | . . . . . 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 2853 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)) |
23 | 22 | ralrimiva 3179 | . . 3 ⊢ (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)) |
24 | 1, 23 | jca 512 | . 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 31462 | . 2 ⊢ (𝜑 → (𝑂 ∈ (toCaraSiga‘𝑀) ↔ (𝑂 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑂)) +𝑒 (𝑀‘(𝑒 ∖ 𝑂))) = (𝑀‘𝑒)))) |
27 | 24, 26 | mpbird 258 | 1 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 +𝑒 cxad 12493 [,]cicc 12729 toCaraSigaccarsg 31458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-xadd 12496 df-icc 12733 df-carsg 31459 |
This theorem is referenced by: carsguni 31465 fiunelcarsg 31473 carsgsiga 31479 |
Copyright terms: Public domain | W3C validator |