Theorem pmap1N 40601

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 2765	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap1.u	. . . 4 ⊢ 1 = (1.‘𝐾)
3	1, 2	op1cl 40019	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4		eqid 2765	. . . 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 40591	. . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
8	3, 7	mpdan 700	. 2 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
9	1, 5	atbase 40123	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
10	1, 4, 2	ople1 40025	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
11	9, 10	sylan2 605	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾) 1 )
12	11	ralrimiva 3159	. . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
13		rabid2 3451	. . 3 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 } ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
14	12, 13	sylibr 237	. 2 ⊢ (𝐾 ∈ OP → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
15	8, 14	eqtr4d 2803	1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418   class class class wbr 5111  ‘cfv 6540  Basecbs 17293  lecple 17341  1.cp1 18502  OPcops 40006  Atomscatm 40097  pmapcpmap 40331

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-lub 18424  df-p1 18504  df-oposet 40010  df-ats 40101  df-pmap 40338

This theorem is used by:  pmapglb2N  40605  pmapglb2xN  40606