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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 2726 | . 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂 | |
3 | caragen0.s | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | 0elpw 5360 | . . 3 ⊢ ∅ ∈ 𝒫 ∪ dom 𝑂 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ∈ 𝒫 ∪ dom 𝑂) |
6 | in0 4396 | . . . . . 6 ⊢ (𝑎 ∩ ∅) = ∅ | |
7 | 6 | fveq2i 6904 | . . . . 5 ⊢ (𝑂‘(𝑎 ∩ ∅)) = (𝑂‘∅) |
8 | dif0 4377 | . . . . . 6 ⊢ (𝑎 ∖ ∅) = 𝑎 | |
9 | 8 | fveq2i 6904 | . . . . 5 ⊢ (𝑂‘(𝑎 ∖ ∅)) = (𝑂‘𝑎) |
10 | 7, 9 | oveq12i 7436 | . . . 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 46118 | . . . . 5 ⊢ (𝜑 → (𝑂‘∅) = 0) |
13 | 12 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘∅) = 0) |
14 | 13 | oveq1d 7439 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘∅) +𝑒 (𝑂‘𝑎)) = (0 +𝑒 (𝑂‘𝑎))) |
15 | iccssxr 13461 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
16 | 1 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas) |
17 | elpwi 4614 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂) | |
18 | 17 | adantl 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂) |
19 | 16, 2, 18 | omecl 46124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ (0[,]+∞)) |
20 | 15, 19 | sselid 3977 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ ℝ*) |
21 | 20 | xaddlidd 44934 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (0 +𝑒 (𝑂‘𝑎)) = (𝑂‘𝑎)) |
22 | 11, 14, 21 | 3eqtrd 2770 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∅)) +𝑒 (𝑂‘(𝑎 ∖ ∅))) = (𝑂‘𝑎)) |
23 | 1, 2, 3, 5, 22 | carageneld 46123 | 1 ⊢ (𝜑 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 𝒫 cpw 4607 ∪ cuni 4913 dom cdm 5682 ‘cfv 6554 (class class class)co 7424 0cc0 11158 +∞cpnf 11295 ℝ*cxr 11297 +𝑒 cxad 13144 [,]cicc 13381 OutMeascome 46110 CaraGenccaragen 46112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-xadd 13147 df-icc 13385 df-ome 46111 df-caragen 46113 |
This theorem is referenced by: caragenfiiuncl 46136 caragenunicl 46145 caragensal 46146 caratheodory 46149 |
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