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Theorem pmaple 38632
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 𝐡 = (Baseβ€˜πΎ)
pmaple.l ≀ = (leβ€˜πΎ)
pmaple.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
pmaple ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (π‘€β€˜π‘‹) βŠ† (π‘€β€˜π‘Œ)))
Proof of Theorem pmaple
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 hlpos 38236 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
2 pmaple.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΎ)
3 eqid 2733 . . . . . . . . . 10 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
42, 3atbase 38159 . . . . . . . . 9 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ 𝐡)
5 pmaple.l . . . . . . . . . . . . . . 15 ≀ = (leβ€˜πΎ)
62, 5postr 18273 . . . . . . . . . . . . . 14 ((𝐾 ∈ Poset ∧ (𝑝 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((𝑝 ≀ 𝑋 ∧ 𝑋 ≀ π‘Œ) β†’ 𝑝 ≀ π‘Œ))
76exp4b 432 . . . . . . . . . . . . 13 (𝐾 ∈ Poset β†’ ((𝑝 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑝 ≀ 𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑝 ≀ π‘Œ))))
873expd 1354 . . . . . . . . . . . 12 (𝐾 ∈ Poset β†’ (𝑝 ∈ 𝐡 β†’ (𝑋 ∈ 𝐡 β†’ (π‘Œ ∈ 𝐡 β†’ (𝑝 ≀ 𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑝 ≀ π‘Œ))))))
98com23 86 . . . . . . . . . . 11 (𝐾 ∈ Poset β†’ (𝑋 ∈ 𝐡 β†’ (𝑝 ∈ 𝐡 β†’ (π‘Œ ∈ 𝐡 β†’ (𝑝 ≀ 𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑝 ≀ π‘Œ))))))
109com34 91 . . . . . . . . . 10 (𝐾 ∈ Poset β†’ (𝑋 ∈ 𝐡 β†’ (π‘Œ ∈ 𝐡 β†’ (𝑝 ∈ 𝐡 β†’ (𝑝 ≀ 𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑝 ≀ π‘Œ))))))
11103imp 1112 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑝 ∈ 𝐡 β†’ (𝑝 ≀ 𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑝 ≀ π‘Œ))))
124, 11syl5 34 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) β†’ (𝑝 ≀ 𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑝 ≀ π‘Œ))))
1312com34 91 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) β†’ (𝑋 ≀ π‘Œ β†’ (𝑝 ≀ 𝑋 β†’ 𝑝 ≀ π‘Œ))))
1413com23 86 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ (𝑝 ∈ (Atomsβ€˜πΎ) β†’ (𝑝 ≀ 𝑋 β†’ 𝑝 ≀ π‘Œ))))
1514ralrimdv 3153 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ βˆ€π‘ ∈ (Atomsβ€˜πΎ)(𝑝 ≀ 𝑋 β†’ 𝑝 ≀ π‘Œ)))
161, 15syl3an1 1164 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ βˆ€π‘ ∈ (Atomsβ€˜πΎ)(𝑝 ≀ 𝑋 β†’ 𝑝 ≀ π‘Œ)))
17 ss2rab 4069 . . . 4 ({𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} ↔ βˆ€π‘ ∈ (Atomsβ€˜πΎ)(𝑝 ≀ 𝑋 β†’ 𝑝 ≀ π‘Œ))
1816, 17syl6ibr 252 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}))
19 hlclat 38228 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
20 ssrab2 4078 . . . . . . . . 9 {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} βŠ† (Atomsβ€˜πΎ)
212, 3atssbase 38160 . . . . . . . . 9 (Atomsβ€˜πΎ) βŠ† 𝐡
2220, 21sstri 3992 . . . . . . . 8 {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} βŠ† 𝐡
23 eqid 2733 . . . . . . . . 9 (lubβ€˜πΎ) = (lubβ€˜πΎ)
242, 5, 23lubss 18466 . . . . . . . 8 ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} βŠ† 𝐡 ∧ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}) β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) ≀ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}))
2522, 24mp3an2 1450 . . . . . . 7 ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}) β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) ≀ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}))
2625ex 414 . . . . . 6 (𝐾 ∈ CLat β†’ ({𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) ≀ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ})))
2719, 26syl 17 . . . . 5 (𝐾 ∈ HL β†’ ({𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) ≀ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ})))
28273ad2ant1 1134 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ({𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) ≀ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ})))
29 hlomcmat 38235 . . . . . . 7 (𝐾 ∈ HL β†’ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
30293ad2ant1 1134 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
31 simp2 1138 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
322, 5, 23, 3atlatmstc 38189 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐡) β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) = 𝑋)
3330, 31, 32syl2anc 585 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) = 𝑋)
34 simp3 1139 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
352, 5, 23, 3atlatmstc 38189 . . . . . 6 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ π‘Œ ∈ 𝐡) β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}) = π‘Œ)
3630, 34, 35syl2anc 585 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}) = π‘Œ)
3733, 36breq12d 5162 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋}) ≀ ((lubβ€˜πΎ)β€˜{𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}) ↔ 𝑋 ≀ π‘Œ))
3828, 37sylibd 238 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ({𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ} β†’ 𝑋 ≀ π‘Œ))
3918, 38impbid 211 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}))
40 pmaple.m . . . . 5 𝑀 = (pmapβ€˜πΎ)
412, 5, 3, 40pmapval 38628 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘€β€˜π‘‹) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋})
42413adant3 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘‹) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋})
432, 5, 3, 40pmapval 38628 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘Œ) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ})
44433adant2 1132 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘Œ) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ})
4542, 44sseq12d 4016 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) βŠ† (π‘€β€˜π‘Œ) ↔ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ 𝑋} βŠ† {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝 ≀ π‘Œ}))
4639, 45bitr4d 282 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (π‘€β€˜π‘‹) βŠ† (π‘€β€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433   βŠ† wss 3949   class class class wbr 5149  β€˜cfv 6544  Basecbs 17144  lecple 17204  Posetcpo 18260  lubclub 18262  CLatccla 18451  OMLcoml 38045  Atomscatm 38133  AtLatcal 38134  HLchlt 38220  pmapcpmap 38368
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-pmap 38375
This theorem is referenced by:  pmap11  38633  hlmod1i  38727  paddunN  38798  pmapojoinN  38839  pl42N  38854
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