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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 | β’ (π β π β OutMeas) |
caragenuncl.2 | β’ π = (CaraGenβπ) |
caragenuncl.3 | β’ (π β πΈ β π) |
caragenuncl.4 | β’ (π β πΉ β π) |
Ref | Expression |
---|---|
caragenuncl | β’ (π β (πΈ βͺ πΉ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenuncl.1 | . 2 β’ (π β π β OutMeas) | |
2 | eqid 2732 | . 2 β’ βͺ dom π = βͺ dom π | |
3 | caragenuncl.2 | . 2 β’ π = (CaraGenβπ) | |
4 | caragenuncl.3 | . . . . 5 β’ (π β πΈ β π) | |
5 | 1, 3, 4, 2 | caragenelss 45203 | . . . 4 β’ (π β πΈ β βͺ dom π) |
6 | caragenuncl.4 | . . . . 5 β’ (π β πΉ β π) | |
7 | 1, 3, 6, 2 | caragenelss 45203 | . . . 4 β’ (π β πΉ β βͺ dom π) |
8 | 5, 7 | unssd 4185 | . . 3 β’ (π β (πΈ βͺ πΉ) β βͺ dom π) |
9 | 1, 2 | unidmex 43722 | . . . . 5 β’ (π β βͺ dom π β V) |
10 | ssexg 5322 | . . . . 5 β’ (((πΈ βͺ πΉ) β βͺ dom π β§ βͺ dom π β V) β (πΈ βͺ πΉ) β V) | |
11 | 8, 9, 10 | syl2anc 584 | . . . 4 β’ (π β (πΈ βͺ πΉ) β V) |
12 | elpwg 4604 | . . . 4 β’ ((πΈ βͺ πΉ) β V β ((πΈ βͺ πΉ) β π« βͺ dom π β (πΈ βͺ πΉ) β βͺ dom π)) | |
13 | 11, 12 | syl 17 | . . 3 β’ (π β ((πΈ βͺ πΉ) β π« βͺ dom π β (πΈ βͺ πΉ) β βͺ dom π)) |
14 | 8, 13 | mpbird 256 | . 2 β’ (π β (πΈ βͺ πΉ) β π« βͺ dom π) |
15 | 1 | adantr 481 | . . 3 β’ ((π β§ π β π« βͺ dom π) β π β OutMeas) |
16 | 4 | adantr 481 | . . 3 β’ ((π β§ π β π« βͺ dom π) β πΈ β π) |
17 | 6 | adantr 481 | . . 3 β’ ((π β§ π β π« βͺ dom π) β πΉ β π) |
18 | elpwi 4608 | . . . 4 β’ (π β π« βͺ dom π β π β βͺ dom π) | |
19 | 18 | adantl 482 | . . 3 β’ ((π β§ π β π« βͺ dom π) β π β βͺ dom π) |
20 | 15, 3, 16, 17, 2, 19 | caragenuncllem 45214 | . 2 β’ ((π β§ π β π« βͺ dom π) β ((πβ(π β© (πΈ βͺ πΉ))) +π (πβ(π β (πΈ βͺ πΉ)))) = (πβπ)) |
21 | 1, 2, 3, 14, 20 | carageneld 45204 | 1 β’ (π β (πΈ βͺ πΉ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 Vcvv 3474 βͺ cun 3945 β wss 3947 π« cpw 4601 βͺ cuni 4907 dom cdm 5675 βcfv 6540 OutMeascome 45191 CaraGenccaragen 45193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-addass 11171 ax-i2m1 11174 ax-rnegex 11177 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-xadd 13089 df-icc 13327 df-ome 45192 df-caragen 45194 |
This theorem is referenced by: caragenfiiuncl 45217 |
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