Mathbox for Glauco Siliprandi |
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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 2818 | . 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 42660 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom 𝑂) |
6 | caragenuncl.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
7 | 1, 3, 6, 2 | caragenelss 42660 | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom 𝑂) |
8 | 5, 7 | unssd 4159 | . . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂) |
9 | 1, 2 | unidmex 41189 | . . . . 5 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
10 | ssexg 5218 | . . . . 5 ⊢ (((𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂 ∧ ∪ dom 𝑂 ∈ V) → (𝐸 ∪ 𝐹) ∈ V) | |
11 | 8, 9, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V) |
12 | elpwg 4541 | . . . 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 258 | . 2 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂) |
15 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas) |
16 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐸 ∈ 𝑆) |
17 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐹 ∈ 𝑆) |
18 | elpwi 4547 | . . . 4 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂) | |
19 | 18 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂) |
20 | 15, 3, 16, 17, 2, 19 | caragenuncllem 42671 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝑎 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝑎)) |
21 | 1, 2, 3, 14, 20 | carageneld 42661 | 1 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 dom cdm 5548 ‘cfv 6348 OutMeascome 42648 CaraGenccaragen 42650 |
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-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-addass 10590 ax-i2m1 10593 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
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-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-le 10669 df-xadd 12496 df-icc 12733 df-ome 42649 df-caragen 42651 |
This theorem is referenced by: caragenfiiuncl 42674 |
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