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Theorem pmap1N 39726
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.‘𝐾)
pmap1.a 𝐴 = (Atoms‘𝐾)
pmap1.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmap1N (𝐾 ∈ OP → (𝑀1 ) = 𝐴)

Proof of Theorem pmap1N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2740 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap1.u . . . 4 1 = (1.‘𝐾)
31, 2op1cl 39143 . . 3 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4 eqid 2740 . . . 4 (le‘𝐾) = (le‘𝐾)
5 pmap1.a . . . 4 𝐴 = (Atoms‘𝐾)
6 pmap1.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 39716 . . 3 ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
83, 7mpdan 686 . 2 (𝐾 ∈ OP → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
91, 5atbase 39247 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
101, 4, 2ople1 39149 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
119, 10sylan2 592 . . . 4 ((𝐾 ∈ OP ∧ 𝑝𝐴) → 𝑝(le‘𝐾) 1 )
1211ralrimiva 3152 . . 3 (𝐾 ∈ OP → ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
13 rabid2 3478 . . 3 (𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 } ↔ ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
1412, 13sylibr 234 . 2 (𝐾 ∈ OP → 𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 })
158, 14eqtr4d 2783 1 (𝐾 ∈ OP → (𝑀1 ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  wral 3067  {crab 3443   class class class wbr 5166  cfv 6575  Basecbs 17260  lecple 17320  1.cp1 18496  OPcops 39130  Atomscatm 39221  pmapcpmap 39456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-lub 18418  df-p1 18498  df-oposet 39134  df-ats 39225  df-pmap 39463
This theorem is referenced by:  pmapglb2N  39730  pmapglb2xN  39731
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