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Theorem pmaple 36884
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 36489 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Poset)
2 pmaple.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
3 eqid 2819 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
42, 3atbase 36412 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
5 pmaple.l . . . . . . . . . . . . . . 15 = (le‘𝐾)
62, 5postr 17555 . . . . . . . . . . . . . 14 ((𝐾 ∈ Poset ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
76exp4b 433 . . . . . . . . . . . . 13 (𝐾 ∈ Poset → ((𝑝𝐵𝑋𝐵𝑌𝐵) → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
873expd 1347 . . . . . . . . . . . 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 1105 . . . . . . . . 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 3186 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
161, 15syl3an1 1157 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌)))
17 ss2rab 4045 . . . 4 ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ↔ ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌))
1816, 17syl6ibr 254 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
19 hlclat 36481 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ CLat)
20 ssrab2 4054 . . . . . . . . 9 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ (Atoms‘𝐾)
212, 3atssbase 36413 . . . . . . . . 9 (Atoms‘𝐾) ⊆ 𝐵
2220, 21sstri 3974 . . . . . . . 8 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ 𝐵
23 eqid 2819 . . . . . . . . 9 (lub‘𝐾) = (lub‘𝐾)
242, 5, 23lubss 17723 . . . . . . . 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 1442 . . . . . . 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 415 . . . . . 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 1127 . . . 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 36488 . . . . . . 7 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
30293ad2ant1 1127 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
31 simp2 1131 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
322, 5, 23, 3atlatmstc 36442 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
3330, 31, 32syl2anc 586 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) = 𝑋)
34 simp3 1132 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
352, 5, 23, 3atlatmstc 36442 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3630, 34, 35syl2anc 586 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) = 𝑌)
3733, 36breq12d 5070 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) ↔ 𝑋 𝑌))
3828, 37sylibd 241 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} → 𝑋 𝑌))
3918, 38impbid 214 . 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 36880 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
42413adant3 1126 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋})
432, 5, 3, 40pmapval 36880 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
44433adant2 1125 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
4542, 44sseq12d 3998 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
4639, 45bitr4d 284 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑀𝑋) ⊆ (𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1081   = wceq 1530  wcel 2107  wral 3136  {crab 3140  wss 3934   class class class wbr 5057  cfv 6348  Basecbs 16475  lecple 16564  Posetcpo 17542  lubclub 17544  CLatccla 17709  OMLcoml 36298  Atomscatm 36386  AtLatcal 36387  HLchlt 36473  pmapcpmap 36620
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36299  df-ol 36301  df-oml 36302  df-covers 36389  df-ats 36390  df-atl 36421  df-cvlat 36445  df-hlat 36474  df-pmap 36627
This theorem is referenced by:  pmap11  36885  hlmod1i  36979  paddunN  37050  pmapojoinN  37091  pl42N  37106
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