Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmaple Structured version   Visualization version   GIF version

Theorem pmaple 37775
Description: The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
pmaple.b 𝐵 = (Base‘𝐾)
pmaple.l = (le‘𝐾)
pmaple.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmaple ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑀𝑋) ⊆ (𝑀𝑌)))

Proof of Theorem pmaple
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 hlpos 37380 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Poset)
2 pmaple.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
3 eqid 2738 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
42, 3atbase 37303 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
5 pmaple.l . . . . . . . . . . . . . . 15 = (le‘𝐾)
62, 5postr 18038 . . . . . . . . . . . . . 14 ((𝐾 ∈ Poset ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
76exp4b 431 . . . . . . . . . . . . 13 (𝐾 ∈ Poset → ((𝑝𝐵𝑋𝐵𝑌𝐵) → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
873expd 1352 . . . . . . . . . . . 12 (𝐾 ∈ Poset → (𝑝𝐵 → (𝑋𝐵 → (𝑌𝐵 → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))))
98com23 86 . . . . . . . . . . 11 (𝐾 ∈ Poset → (𝑋𝐵 → (𝑝𝐵 → (𝑌𝐵 → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))))
109com34 91 . . . . . . . . . 10 (𝐾 ∈ Poset → (𝑋𝐵 → (𝑌𝐵 → (𝑝𝐵 → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))))
11103imp 1110 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑝𝐵 → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
124, 11syl5 34 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
1312com34 91 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑋 𝑌 → (𝑝 𝑋𝑝 𝑌))))
1413com23 86 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 𝑋𝑝 𝑌))))
1514ralrimdv 3105 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
161, 15syl3an1 1162 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
17 ss2rab 4004 . . . 4 ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ↔ ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌))
1816, 17syl6ibr 251 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
19 hlclat 37372 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ CLat)
20 ssrab2 4013 . . . . . . . . 9 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ (Atoms‘𝐾)
212, 3atssbase 37304 . . . . . . . . 9 (Atoms‘𝐾) ⊆ 𝐵
2220, 21sstri 3930 . . . . . . . 8 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ 𝐵
23 eqid 2738 . . . . . . . . 9 (lub‘𝐾) = (lub‘𝐾)
242, 5, 23lubss 18231 . . . . . . . 8 ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ 𝐵 ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
2522, 24mp3an2 1448 . . . . . . 7 ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
2625ex 413 . . . . . 6 (𝐾 ∈ CLat → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})))
2719, 26syl 17 . . . . 5 (𝐾 ∈ HL → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})))
28273ad2ant1 1132 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})))
29 hlomcmat 37379 . . . . . . 7 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
30293ad2ant1 1132 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
31 simp2 1136 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
322, 5, 23, 3atlatmstc 37333 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
3330, 31, 32syl2anc 584 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
34 simp3 1137 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
352, 5, 23, 3atlatmstc 37333 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3630, 34, 35syl2anc 584 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3733, 36breq12d 5087 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) ↔ 𝑋 𝑌))
3828, 37sylibd 238 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} → 𝑋 𝑌))
3918, 38impbid 211 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
40 pmaple.m . . . . 5 𝑀 = (pmap‘𝐾)
412, 5, 3, 40pmapval 37771 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
42413adant3 1131 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
432, 5, 3, 40pmapval 37771 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
44433adant2 1130 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
4542, 44sseq12d 3954 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
4639, 45bitr4d 281 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑀𝑋) ⊆ (𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086   = wceq 1539  wcel 2106  wral 3064  {crab 3068  wss 3887   class class class wbr 5074  cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025  lubclub 18027  CLatccla 18216  OMLcoml 37189  Atomscatm 37277  AtLatcal 37278  HLchlt 37364  pmapcpmap 37511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-pmap 37518
This theorem is referenced by:  pmap11  37776  hlmod1i  37870  paddunN  37941  pmapojoinN  37982  pl42N  37997
  Copyright terms: Public domain W3C validator