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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > carageneld | Structured version Visualization version GIF version |
Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
carageneld.o | β’ (π β π β OutMeas) |
carageneld.x | β’ π = βͺ dom π |
carageneld.s | β’ π = (CaraGenβπ) |
carageneld.e | β’ (π β πΈ β π« π) |
carageneld.a | β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)) |
Ref | Expression |
---|---|
carageneld | β’ (π β πΈ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carageneld.e | . . . 4 β’ (π β πΈ β π« π) | |
2 | carageneld.x | . . . . 5 β’ π = βͺ dom π | |
3 | 2 | pweqi 4617 | . . . 4 β’ π« π = π« βͺ dom π |
4 | 1, 3 | eleqtrdi 2843 | . . 3 β’ (π β πΈ β π« βͺ dom π) |
5 | simpl 483 | . . . . 5 β’ ((π β§ π β π« βͺ dom π) β π) | |
6 | 3 | eleq2i 2825 | . . . . . . . 8 β’ (π β π« π β π β π« βͺ dom π) |
7 | 6 | bicomi 223 | . . . . . . 7 β’ (π β π« βͺ dom π β π β π« π) |
8 | 7 | biimpi 215 | . . . . . 6 β’ (π β π« βͺ dom π β π β π« π) |
9 | 8 | adantl 482 | . . . . 5 β’ ((π β§ π β π« βͺ dom π) β π β π« π) |
10 | carageneld.a | . . . . 5 β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)) | |
11 | 5, 9, 10 | syl2anc 584 | . . . 4 β’ ((π β§ π β π« βͺ dom π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)) |
12 | 11 | ralrimiva 3146 | . . 3 β’ (π β βπ β π« βͺ dom π((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)) |
13 | 4, 12 | jca 512 | . 2 β’ (π β (πΈ β π« βͺ dom π β§ βπ β π« βͺ dom π((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ))) |
14 | carageneld.o | . . 3 β’ (π β π β OutMeas) | |
15 | carageneld.s | . . 3 β’ π = (CaraGenβπ) | |
16 | 14, 15 | caragenel 45197 | . 2 β’ (π β (πΈ β π β (πΈ β π« βͺ dom π β§ βπ β π« βͺ dom π((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)))) |
17 | 13, 16 | mpbird 256 | 1 β’ (π β πΈ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β cdif 3944 β© cin 3946 π« cpw 4601 βͺ cuni 4907 dom cdm 5675 βcfv 6540 (class class class)co 7405 +π cxad 13086 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 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-caragen 45194 |
This theorem is referenced by: caragen0 45208 caragenunidm 45210 caragenuncl 45215 caragendifcl 45216 carageniuncl 45225 caragenel2d 45234 |
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