![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pointsetN | Structured version Visualization version GIF version |
Description: The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pointset.a | β’ π΄ = (AtomsβπΎ) |
pointset.p | β’ π = (PointsβπΎ) |
Ref | Expression |
---|---|
pointsetN | β’ (πΎ β π΅ β π = {π β£ βπ β π΄ π = {π}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3490 | . 2 β’ (πΎ β π΅ β πΎ β V) | |
2 | pointset.p | . . 3 β’ π = (PointsβπΎ) | |
3 | fveq2 6897 | . . . . . . 7 β’ (π = πΎ β (Atomsβπ) = (AtomsβπΎ)) | |
4 | pointset.a | . . . . . . 7 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | eqtr4di 2786 | . . . . . 6 β’ (π = πΎ β (Atomsβπ) = π΄) |
6 | 5 | rexeqdv 3323 | . . . . 5 β’ (π = πΎ β (βπ β (Atomsβπ)π = {π} β βπ β π΄ π = {π})) |
7 | 6 | abbidv 2797 | . . . 4 β’ (π = πΎ β {π β£ βπ β (Atomsβπ)π = {π}} = {π β£ βπ β π΄ π = {π}}) |
8 | df-pointsN 38975 | . . . 4 β’ Points = (π β V β¦ {π β£ βπ β (Atomsβπ)π = {π}}) | |
9 | 4 | fvexi 6911 | . . . . 5 β’ π΄ β V |
10 | 9 | abrexex 7966 | . . . 4 β’ {π β£ βπ β π΄ π = {π}} β V |
11 | 7, 8, 10 | fvmpt 7005 | . . 3 β’ (πΎ β V β (PointsβπΎ) = {π β£ βπ β π΄ π = {π}}) |
12 | 2, 11 | eqtrid 2780 | . 2 β’ (πΎ β V β π = {π β£ βπ β π΄ π = {π}}) |
13 | 1, 12 | syl 17 | 1 β’ (πΎ β π΅ β π = {π β£ βπ β π΄ π = {π}}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 {cab 2705 βwrex 3067 Vcvv 3471 {csn 4629 βcfv 6548 Atomscatm 38735 PointscpointsN 38968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-pointsN 38975 |
This theorem is referenced by: ispointN 39215 |
Copyright terms: Public domain | W3C validator |