Theorem pmap1N 39754

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 → (𝑀‘ 1 ) = 𝐴)

Proof of Theorem pmap1N

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap1.u	. . . 4 ⊢ 1 = (1.‘𝐾)
3	1, 2	op1cl 39171	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4		eqid 2729	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
5		pmap1.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		pmap1.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
7	1, 4, 5, 6	pmapval 39744	. . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
8	3, 7	mpdan 687	. 2 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
9	1, 5	atbase 39275	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
10	1, 4, 2	ople1 39177	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
11	9, 10	sylan2 593	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾) 1 )
12	11	ralrimiva 3125	. . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
13		rabid2 3436	. . 3 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 } ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
14	12, 13	sylibr 234	. 2 ⊢ (𝐾 ∈ OP → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
15	8, 14	eqtr4d 2767	1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402   class class class wbr 5102  ‘cfv 6499  Basecbs 17155  lecple 17203  1.cp1 18363  OPcops 39158  Atomscatm 39249  pmapcpmap 39484

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-lub 18285  df-p1 18365  df-oposet 39162  df-ats 39253  df-pmap 39491

This theorem is referenced by:  pmapglb2N  39758  pmapglb2xN  39759