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Theorem pmap1N 38941
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.)
Hypotheses
Ref Expression
pmap1.u 1 = (1.β€˜πΎ)
pmap1.a 𝐴 = (Atomsβ€˜πΎ)
pmap1.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
pmap1N (𝐾 ∈ OP β†’ (π‘€β€˜ 1 ) = 𝐴)

Proof of Theorem pmap1N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 pmap1.u . . . 4 1 = (1.β€˜πΎ)
31, 2op1cl 38358 . . 3 (𝐾 ∈ OP β†’ 1 ∈ (Baseβ€˜πΎ))
4 eqid 2730 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
5 pmap1.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
6 pmap1.m . . . 4 𝑀 = (pmapβ€˜πΎ)
71, 4, 5, 6pmapval 38931 . . 3 ((𝐾 ∈ OP ∧ 1 ∈ (Baseβ€˜πΎ)) β†’ (π‘€β€˜ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ) 1 })
83, 7mpdan 683 . 2 (𝐾 ∈ OP β†’ (π‘€β€˜ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ) 1 })
91, 5atbase 38462 . . . . 5 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ (Baseβ€˜πΎ))
101, 4, 2ople1 38364 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ 𝑝(leβ€˜πΎ) 1 )
119, 10sylan2 591 . . . 4 ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) β†’ 𝑝(leβ€˜πΎ) 1 )
1211ralrimiva 3144 . . 3 (𝐾 ∈ OP β†’ βˆ€π‘ ∈ 𝐴 𝑝(leβ€˜πΎ) 1 )
13 rabid2 3462 . . 3 (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ) 1 } ↔ βˆ€π‘ ∈ 𝐴 𝑝(leβ€˜πΎ) 1 )
1412, 13sylibr 233 . 2 (𝐾 ∈ OP β†’ 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ) 1 })
158, 14eqtr4d 2773 1 (𝐾 ∈ OP β†’ (π‘€β€˜ 1 ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430   class class class wbr 5147  β€˜cfv 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
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