![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenuncl | Structured version Visualization version GIF version |
Description: The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenuncl.1 | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenuncl.2 | ⊢ 𝑆 = (CaraGen‘𝑂) |
caragenuncl.3 | ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
caragenuncl.4 | ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
Ref | Expression |
---|---|
caragenuncl | ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenuncl.1 | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | eqid 2735 | . 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂 | |
3 | caragenuncl.2 | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | caragenuncl.3 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | |
5 | 1, 3, 4, 2 | caragenelss 46457 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom 𝑂) |
6 | caragenuncl.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
7 | 1, 3, 6, 2 | caragenelss 46457 | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom 𝑂) |
8 | 5, 7 | unssd 4202 | . . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂) |
9 | 1, 2 | unidmex 44990 | . . . . 5 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
10 | ssexg 5329 | . . . . 5 ⊢ (((𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂 ∧ ∪ dom 𝑂 ∈ V) → (𝐸 ∪ 𝐹) ∈ V) | |
11 | 8, 9, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V) |
12 | elpwg 4608 | . . . 4 ⊢ ((𝐸 ∪ 𝐹) ∈ V → ((𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂 ↔ (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂 ↔ (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂)) |
14 | 8, 13 | mpbird 257 | . 2 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂) |
15 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas) |
16 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐸 ∈ 𝑆) |
17 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐹 ∈ 𝑆) |
18 | elpwi 4612 | . . . 4 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂) | |
19 | 18 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂) |
20 | 15, 3, 16, 17, 2, 19 | caragenuncllem 46468 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝑎 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝑎)) |
21 | 1, 2, 3, 14, 20 | carageneld 46458 | 1 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 dom cdm 5689 ‘cfv 6563 OutMeascome 46445 CaraGenccaragen 46447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-addass 11218 ax-i2m1 11221 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-xadd 13153 df-icc 13391 df-ome 46446 df-caragen 46448 |
This theorem is referenced by: caragenfiiuncl 46471 |
Copyright terms: Public domain | W3C validator |