![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pmap1N | Structured version Visualization version GIF version |
Description: Value of the projective map of a Hilbert lattice at lattice unity. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pmap1.u | β’ 1 = (1.βπΎ) |
pmap1.a | β’ π΄ = (AtomsβπΎ) |
pmap1.m | β’ π = (pmapβπΎ) |
Ref | Expression |
---|---|
pmap1N | β’ (πΎ β OP β (πβ 1 ) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | pmap1.u | . . . 4 β’ 1 = (1.βπΎ) | |
3 | 1, 2 | op1cl 38358 | . . 3 β’ (πΎ β OP β 1 β (BaseβπΎ)) |
4 | eqid 2730 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
5 | pmap1.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | pmap1.m | . . . 4 β’ π = (pmapβπΎ) | |
7 | 1, 4, 5, 6 | pmapval 38931 | . . 3 β’ ((πΎ β OP β§ 1 β (BaseβπΎ)) β (πβ 1 ) = {π β π΄ β£ π(leβπΎ) 1 }) |
8 | 3, 7 | mpdan 683 | . 2 β’ (πΎ β OP β (πβ 1 ) = {π β π΄ β£ π(leβπΎ) 1 }) |
9 | 1, 5 | atbase 38462 | . . . . 5 β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 1, 4, 2 | ople1 38364 | . . . . 5 β’ ((πΎ β OP β§ π β (BaseβπΎ)) β π(leβπΎ) 1 ) |
11 | 9, 10 | sylan2 591 | . . . 4 β’ ((πΎ β OP β§ π β π΄) β π(leβπΎ) 1 ) |
12 | 11 | ralrimiva 3144 | . . 3 β’ (πΎ β OP β βπ β π΄ π(leβπΎ) 1 ) |
13 | rabid2 3462 | . . 3 β’ (π΄ = {π β π΄ β£ π(leβπΎ) 1 } β βπ β π΄ π(leβπΎ) 1 ) | |
14 | 12, 13 | sylibr 233 | . 2 β’ (πΎ β OP β π΄ = {π β π΄ β£ π(leβπΎ) 1 }) |
15 | 8, 14 | eqtr4d 2773 | 1 β’ (πΎ β OP β (πβ 1 ) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βwral 3059 {crab 3430 class class class wbr 5147 βcfv 6542 Basecbs 17148 lecple 17208 1.cp1 18381 OPcops 38345 Atomscatm 38436 pmapcpmap 38671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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-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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-lub 18303 df-p1 18383 df-oposet 38349 df-ats 38440 df-pmap 38678 |
This theorem is referenced by: pmapglb2N 38945 pmapglb2xN 38946 |
Copyright terms: Public domain | W3C validator |