Theorem pmap1N 40172

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 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap1.u	. . . 4 ⊢ 1 = (1.‘𝐾)
3	1, 2	op1cl 39590	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4		eqid 2737	. . . 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 40162	. . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
8	3, 7	mpdan 688	. 2 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
9	1, 5	atbase 39694	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
10	1, 4, 2	ople1 39596	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
11	9, 10	sylan2 594	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾) 1 )
12	11	ralrimiva 3130	. . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
13		rabid2 3434	. . 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 2775	1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   class class class wbr 5100  ‘cfv 6502  Basecbs 17150  lecple 17198  1.cp1 18359  OPcops 39577  Atomscatm 39668  pmapcpmap 39902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-lub 18281  df-p1 18361  df-oposet 39581  df-ats 39672  df-pmap 39909

This theorem is referenced by:  pmapglb2N  40176  pmapglb2xN  40177