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Theorem pmaple 39743
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 39347 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Poset)
2 pmaple.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
3 eqid 2734 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
42, 3atbase 39270 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
5 pmaple.l . . . . . . . . . . . . . . 15 = (le‘𝐾)
62, 5postr 18377 . . . . . . . . . . . . . 14 ((𝐾 ∈ Poset ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
76exp4b 430 . . . . . . . . . . . . 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 3149 . . . . 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 4080 . . . 4 ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ↔ ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋𝑝 𝑌))
1816, 17imbitrrdi 252 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}))
19 hlclat 39339 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ CLat)
20 ssrab2 4089 . . . . . . . . 9 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ (Atoms‘𝐾)
212, 3atssbase 39271 . . . . . . . . 9 (Atoms‘𝐾) ⊆ 𝐵
2220, 21sstri 4004 . . . . . . . 8 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} ⊆ 𝐵
23 eqid 2734 . . . . . . . . 9 (lub‘𝐾) = (lub‘𝐾)
242, 5, 23lubss 18570 . . . . . . . 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 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 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 39346 . . . . . . 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 39300 . . . . . 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 39300 . . . . . 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 5160 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋}) ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌}) ↔ 𝑋 𝑌))
3828, 37sylibd 239 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌} → 𝑋 𝑌))
3918, 38impbid 212 . 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 39739 . . . 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 39739 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
44433adant2 1130 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 𝑌})
4542, 44sseq12d 4028 . 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 206  w3a 1086   = wceq 1536  wcel 2105  wral 3058  {crab 3432  wss 3962   class class class wbr 5147  cfv 6562  Basecbs 17244  lecple 17304  Posetcpo 18364  lubclub 18366  CLatccla 18555  OMLcoml 39156  Atomscatm 39244  AtLatcal 39245  HLchlt 39331  pmapcpmap 39479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-lat 18489  df-clat 18556  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-pmap 39486
This theorem is referenced by:  pmap11  39744  hlmod1i  39838  paddunN  39909  pmapojoinN  39950  pl42N  39965
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