![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 4614 | . . . 4 β’ π« π = π« βͺ dom π |
4 | 1, 3 | eleqtrdi 2835 | . . 3 β’ (π β πΈ β π« βͺ dom π) |
5 | simpl 481 | . . . . 5 β’ ((π β§ π β π« βͺ dom π) β π) | |
6 | 3 | eleq2i 2817 | . . . . . . . 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 480 | . . . . 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 582 | . . . 4 β’ ((π β§ π β π« βͺ dom π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)) |
12 | 11 | ralrimiva 3136 | . . 3 β’ (π β βπ β π« βͺ dom π((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)) |
13 | 4, 12 | jca 510 | . 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 45945 | . 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 394 = wceq 1533 β wcel 2098 βwral 3051 β cdif 3937 β© cin 3939 π« cpw 4598 βͺ cuni 4903 dom cdm 5672 βcfv 6542 (class class class)co 7415 +π cxad 13120 OutMeascome 45939 CaraGenccaragen 45941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7418 df-caragen 45942 |
This theorem is referenced by: caragen0 45956 caragenunidm 45958 caragenuncl 45963 caragendifcl 45964 carageniuncl 45973 caragenel2d 45982 |
Copyright terms: Public domain | W3C validator |