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Theorem pmapsub 36386
Description: The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
pmapsub.b 𝐵 = (Base‘𝐾)
pmapsub.s 𝑆 = (PSubSp‘𝐾)
pmapsub.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmapsub ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)

Proof of Theorem pmapsub
Dummy variables 𝑞 𝑝 𝑟 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmapsub.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2771 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2771 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
4 pmapsub.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 36375 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6 breq1 4928 . . . . . . . . . . . . . 14 (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋𝑝(le‘𝐾)𝑋))
76elrab 3588 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
81, 3atbase 35907 . . . . . . . . . . . . . 14 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
98anim1i 606 . . . . . . . . . . . . 13 ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝𝐵𝑝(le‘𝐾)𝑋))
107, 9sylbi 209 . . . . . . . . . . . 12 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑝𝐵𝑝(le‘𝐾)𝑋))
11 breq1 4928 . . . . . . . . . . . . . 14 (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))
1211elrab 3588 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
131, 3atbase 35907 . . . . . . . . . . . . . 14 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
1413anim1i 606 . . . . . . . . . . . . 13 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1512, 14sylbi 209 . . . . . . . . . . . 12 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1610, 15anim12i 604 . . . . . . . . . . 11 ((𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}) → ((𝑝𝐵𝑝(le‘𝐾)𝑋) ∧ (𝑞𝐵𝑞(le‘𝐾)𝑋)))
17 an4 644 . . . . . . . . . . 11 (((𝑝𝐵𝑝(le‘𝐾)𝑋) ∧ (𝑞𝐵𝑞(le‘𝐾)𝑋)) ↔ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋)))
1816, 17sylib 210 . . . . . . . . . 10 ((𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}) → ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋)))
1918anim2i 608 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) → ((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))))
201, 3atbase 35907 . . . . . . . . 9 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
21 eqid 2771 . . . . . . . . . . . . . . . . 17 (join‘𝐾) = (join‘𝐾)
221, 2, 21latjle12 17542 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) ↔ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
2322biimpd 221 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
24233exp2 1335 . . . . . . . . . . . . . 14 (𝐾 ∈ Lat → (𝑝𝐵 → (𝑞𝐵 → (𝑋𝐵 → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)))))
2524impd 402 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → ((𝑝𝐵𝑞𝐵) → (𝑋𝐵 → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))))
2625com23 86 . . . . . . . . . . . 12 (𝐾 ∈ Lat → (𝑋𝐵 → ((𝑝𝐵𝑞𝐵) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))))
2726imp43 420 . . . . . . . . . . 11 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
2827adantr 473 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
291, 21latjcl 17531 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30293expib 1103 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → ((𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
311, 2lattr 17536 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑟𝐵 ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵𝑋𝐵)) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
32313exp2 1335 . . . . . . . . . . . . . 14 (𝐾 ∈ Lat → (𝑟𝐵 → ((𝑝(join‘𝐾)𝑞) ∈ 𝐵 → (𝑋𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
3332com24 95 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → (𝑋𝐵 → ((𝑝(join‘𝐾)𝑞) ∈ 𝐵 → (𝑟𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
3430, 33syl5d 73 . . . . . . . . . . . 12 (𝐾 ∈ Lat → (𝑋𝐵 → ((𝑝𝐵𝑞𝐵) → (𝑟𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
3534imp41 418 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝𝐵𝑞𝐵)) ∧ 𝑟𝐵) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
3635adantlrr 709 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
3728, 36mpan2d 682 . . . . . . . . 9 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)𝑋))
3819, 20, 37syl2an 587 . . . . . . . 8 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)𝑋))
39 simpr 477 . . . . . . . 8 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Atoms‘𝐾))
4038, 39jctild 518 . . . . . . 7 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋)))
41 breq1 4928 . . . . . . . 8 (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋𝑟(le‘𝐾)𝑋))
4241elrab 3588 . . . . . . 7 (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋))
4340, 42syl6ibr 244 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
4443ralrimiva 3125 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) → ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
4544ralrimivva 3134 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
46 ssrab2 3939 . . . 4 {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾)
4745, 46jctil 512 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})))
48 pmapsub.s . . . . 5 𝑆 = (PSubSp‘𝐾)
492, 21, 3, 48ispsubsp 36363 . . . 4 (𝐾 ∈ Lat → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆 ↔ ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))))
5049adantr 473 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆 ↔ ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))))
5147, 50mpbird 249 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆)
525, 51eqeltrd 2859 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wral 3081  {crab 3085  wss 3822   class class class wbr 4925  cfv 6185  (class class class)co 6974  Basecbs 16337  lecple 16426  joincjn 17424  Latclat 17525  Atomscatm 35881  PSubSpcpsubsp 36114  pmapcpmap 36115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-poset 17426  df-lub 17454  df-glb 17455  df-join 17456  df-meet 17457  df-lat 17526  df-ats 35885  df-psubsp 36121  df-pmap 36122
This theorem is referenced by:  hlmod1i  36474  polsubN  36525  pl42lem4N  36600
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