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Theorem poml5N 39427
Description: Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
poml4.a 𝐴 = (Atomsβ€˜πΎ)
poml4.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
poml5N ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))) ∩ ( βŠ₯ β€˜π‘Œ)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))

Proof of Theorem poml5N
StepHypRef Expression
1 simp1 1134 . . 3 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝐾 ∈ HL)
2 simp3 1136 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ))
3 poml4.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
4 poml4.p . . . . . 6 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
53, 4polssatN 39381 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴)
653adant3 1130 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴)
72, 6sstrd 3990 . . 3 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝑋 βŠ† 𝐴)
81, 7, 63jca 1126 . 2 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴))
93, 43polN 39389 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) = ( βŠ₯ β€˜π‘Œ))
1093adant3 1130 . . 3 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) = ( βŠ₯ β€˜π‘Œ))
112, 10jca 511 . 2 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) = ( βŠ₯ β€˜π‘Œ)))
123, 4poml4N 39426 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴) β†’ ((𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) = ( βŠ₯ β€˜π‘Œ)) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))) ∩ ( βŠ₯ β€˜π‘Œ)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
138, 11, 12sylc 65 1 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))) ∩ ( βŠ₯ β€˜π‘Œ)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ∩ cin 3946   βŠ† wss 3947  β€˜cfv 6548  Atomscatm 38735  HLchlt 38822  βŠ₯𝑃cpolN 39375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-proset 18287  df-poset 18305  df-plt 18322  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-p1 18418  df-lat 18424  df-clat 18491  df-oposet 38648  df-ol 38650  df-oml 38651  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823  df-psubsp 38976  df-pmap 38977  df-polarityN 39376
This theorem is referenced by:  osumcllem3N  39431
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