Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragen0 | Structured version Visualization version GIF version |
Description: The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragen0.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragen0.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
Ref | Expression |
---|---|
caragen0 | ⊢ (𝜑 → ∅ ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragen0.o | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | eqid 2823 | . 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂 | |
3 | caragen0.s | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | 0elpw 5258 | . . 3 ⊢ ∅ ∈ 𝒫 ∪ dom 𝑂 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ∈ 𝒫 ∪ dom 𝑂) |
6 | in0 4347 | . . . . . 6 ⊢ (𝑎 ∩ ∅) = ∅ | |
7 | 6 | fveq2i 6675 | . . . . 5 ⊢ (𝑂‘(𝑎 ∩ ∅)) = (𝑂‘∅) |
8 | dif0 4334 | . . . . . 6 ⊢ (𝑎 ∖ ∅) = 𝑎 | |
9 | 8 | fveq2i 6675 | . . . . 5 ⊢ (𝑂‘(𝑎 ∖ ∅)) = (𝑂‘𝑎) |
10 | 7, 9 | oveq12i 7170 | . . . 4 ⊢ ((𝑂‘(𝑎 ∩ ∅)) +𝑒 (𝑂‘(𝑎 ∖ ∅))) = ((𝑂‘∅) +𝑒 (𝑂‘𝑎)) |
11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∅)) +𝑒 (𝑂‘(𝑎 ∖ ∅))) = ((𝑂‘∅) +𝑒 (𝑂‘𝑎))) |
12 | 1 | ome0 42786 | . . . . 5 ⊢ (𝜑 → (𝑂‘∅) = 0) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘∅) = 0) |
14 | 13 | oveq1d 7173 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘∅) +𝑒 (𝑂‘𝑎)) = (0 +𝑒 (𝑂‘𝑎))) |
15 | iccssxr 12822 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
16 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas) |
17 | elpwi 4550 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂) | |
18 | 17 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂) |
19 | 16, 2, 18 | omecl 42792 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ (0[,]+∞)) |
20 | 15, 19 | sseldi 3967 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ ℝ*) |
21 | 20 | xaddid2d 41594 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (0 +𝑒 (𝑂‘𝑎)) = (𝑂‘𝑎)) |
22 | 11, 14, 21 | 3eqtrd 2862 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∅)) +𝑒 (𝑂‘(𝑎 ∖ ∅))) = (𝑂‘𝑎)) |
23 | 1, 2, 3, 5, 22 | carageneld 42791 | 1 ⊢ (𝜑 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ∪ cuni 4840 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 ℝ*cxr 10676 +𝑒 cxad 12508 [,]cicc 12744 OutMeascome 42778 CaraGenccaragen 42780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-xadd 12511 df-icc 12748 df-ome 42779 df-caragen 42781 |
This theorem is referenced by: caragenfiiuncl 42804 caragenunicl 42813 caragensal 42814 caratheodory 42817 |
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