Theorem pmap1N 40213

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 2736	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap1.u	. . . 4 ⊢ 1 = (1.‘𝐾)
3	1, 2	op1cl 39631	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4		eqid 2736	. . . 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 40203	. . 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 39735	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
10	1, 4, 2	ople1 39637	. . . . 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 3129	. . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
13		rabid2 3422	. . 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 2774	1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389   class class class wbr 5085  ‘cfv 6498  Basecbs 17179  lecple 17227  1.cp1 18388  OPcops 39618  Atomscatm 39709  pmapcpmap 39943

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-lub 18310  df-p1 18390  df-oposet 39622  df-ats 39713  df-pmap 39950

This theorem is referenced by:  pmapglb2N  40217  pmapglb2xN  40218