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Theorem pmaple 39122
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 38726 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Poset)
2 pmaple.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
3 eqid 2724 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
42, 3atbase 38649 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
5 pmaple.l . . . . . . . . . . . . . . 15 = (le‘𝐾)
62, 5postr 18275 . . . . . . . . . . . . . 14 ((𝐾 ∈ Poset ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
76exp4b 430 . . . . . . . . . . . . 13 (𝐾 ∈ Poset → ((𝑝𝐵𝑋𝐵𝑌𝐵) → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
873expd 1350 . . . . . . . . . . . 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 1108 . . . . . . . . 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 3144 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
161, 15syl3an1 1160 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
17 ss2rab 4060 . . . 4 ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ↔ ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌))
1816, 17imbitrrdi 251 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
19 hlclat 38718 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ CLat)
20 ssrab2 4069 . . . . . . . . 9 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ (Atoms‘𝐾)
212, 3atssbase 38650 . . . . . . . . 9 (Atoms‘𝐾) ⊆ 𝐵
2220, 21sstri 3983 . . . . . . . 8 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ 𝐵
23 eqid 2724 . . . . . . . . 9 (lub‘𝐾) = (lub‘𝐾)
242, 5, 23lubss 18468 . . . . . . . 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 1445 . . . . . . 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 412 . . . . . 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 1130 . . . 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 38725 . . . . . . 7 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
30293ad2ant1 1130 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
31 simp2 1134 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
322, 5, 23, 3atlatmstc 38679 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
3330, 31, 32syl2anc 583 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
34 simp3 1135 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
352, 5, 23, 3atlatmstc 38679 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3630, 34, 35syl2anc 583 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3733, 36breq12d 5151 . . . 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 39118 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
42413adant3 1129 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
432, 5, 3, 40pmapval 39118 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
44433adant2 1128 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
4542, 44sseq12d 4007 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
4639, 45bitr4d 282 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑀𝑋) ⊆ (𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  wral 3053  {crab 3424  wss 3940   class class class wbr 5138  cfv 6533  Basecbs 17143  lecple 17203  Posetcpo 18262  lubclub 18264  CLatccla 18453  OMLcoml 38535  Atomscatm 38623  AtLatcal 38624  HLchlt 38710  pmapcpmap 38858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18387  df-clat 18454  df-oposet 38536  df-ol 38538  df-oml 38539  df-covers 38626  df-ats 38627  df-atl 38658  df-cvlat 38682  df-hlat 38711  df-pmap 38865
This theorem is referenced by:  pmap11  39123  hlmod1i  39217  paddunN  39288  pmapojoinN  39329  pl42N  39344
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