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Theorem pmaple 35836
 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 35441 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Poset)
2 pmaple.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
3 eqid 2825 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
42, 3atbase 35364 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
5 pmaple.l . . . . . . . . . . . . . . 15 = (le‘𝐾)
62, 5postr 17306 . . . . . . . . . . . . . 14 ((𝐾 ∈ Poset ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
76exp4b 423 . . . . . . . . . . . . 13 (𝐾 ∈ Poset → ((𝑝𝐵𝑋𝐵𝑌𝐵) → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
873expd 1468 . . . . . . . . . . . 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 1143 . . . . . . . . 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 3177 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
161, 15syl3an1 1208 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
17 ss2rab 3903 . . . 4 ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ↔ ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌))
1816, 17syl6ibr 244 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
19 hlclat 35433 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ CLat)
20 ssrab2 3912 . . . . . . . . 9 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ (Atoms‘𝐾)
212, 3atssbase 35365 . . . . . . . . 9 (Atoms‘𝐾) ⊆ 𝐵
2220, 21sstri 3836 . . . . . . . 8 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ 𝐵
23 eqid 2825 . . . . . . . . 9 (lub‘𝐾) = (lub‘𝐾)
242, 5, 23lubss 17474 . . . . . . . 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 1579 . . . . . . 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 403 . . . . . 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 1169 . . . 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 35440 . . . . . . 7 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
30293ad2ant1 1169 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
31 simp2 1173 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
322, 5, 23, 3atlatmstc 35394 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
3330, 31, 32syl2anc 581 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
34 simp3 1174 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
352, 5, 23, 3atlatmstc 35394 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3630, 34, 35syl2anc 581 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3733, 36breq12d 4886 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) ↔ 𝑋 𝑌))
3828, 37sylibd 231 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} → 𝑋 𝑌))
3918, 38impbid 204 . 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 35832 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
42413adant3 1168 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
432, 5, 3, 40pmapval 35832 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
44433adant2 1167 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
4542, 44sseq12d 3859 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
4639, 45bitr4d 274 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑀𝑋) ⊆ (𝑀𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3117  {crab 3121   ⊆ wss 3798   class class class wbr 4873  ‘cfv 6123  Basecbs 16222  lecple 16312  Posetcpo 17293  lubclub 17295  CLatccla 17460  OMLcoml 35250  Atomscatm 35338  AtLatcal 35339  HLchlt 35425  pmapcpmap 35572 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35251  df-ol 35253  df-oml 35254  df-covers 35341  df-ats 35342  df-atl 35373  df-cvlat 35397  df-hlat 35426  df-pmap 35579 This theorem is referenced by:  pmap11  35837  hlmod1i  35931  paddunN  36002  pmapojoinN  36043  pl42N  36058
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