Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poml6N Structured version   Visualization version   GIF version

Theorem poml6N 37083
Description: Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
poml6.c 𝐶 = (PSubCl‘𝐾)
poml6.p = (⊥𝑃𝐾)
Assertion
Ref Expression
poml6N (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)

Proof of Theorem poml6N
StepHypRef Expression
1 simpl1 1186 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → 𝐾 ∈ HL)
2 simpl2 1187 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → 𝑋𝐶)
3 eqid 2819 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
4 poml6.c . . . . 5 𝐶 = (PSubCl‘𝐾)
53, 4psubclssatN 37069 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
61, 2, 5syl2anc 586 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → 𝑋 ⊆ (Atoms‘𝐾))
7 simpl3 1188 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → 𝑌𝐶)
83, 4psubclssatN 37069 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐶) → 𝑌 ⊆ (Atoms‘𝐾))
91, 7, 8syl2anc 586 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → 𝑌 ⊆ (Atoms‘𝐾))
10 simpr 487 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → 𝑋𝑌)
11 poml6.p . . . . 5 = (⊥𝑃𝐾)
1211, 4psubcli2N 37067 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐶) → ( ‘( 𝑌)) = 𝑌)
131, 7, 12syl2anc 586 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → ( ‘( 𝑌)) = 𝑌)
143, 11poml4N 37081 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → ((𝑋𝑌 ∧ ( ‘( 𝑌)) = 𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ‘( 𝑋))))
1514imp 409 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) ∧ (𝑋𝑌 ∧ ( ‘( 𝑌)) = 𝑌)) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ‘( 𝑋)))
161, 6, 9, 10, 13, 15syl32anc 1373 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ‘( 𝑋)))
1711, 4psubcli2N 37067 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)
181, 2, 17syl2anc 586 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → ( ‘( 𝑋)) = 𝑋)
1916, 18eqtrd 2854 1 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1082   = wceq 1531  wcel 2108  cin 3933  wss 3934  cfv 6348  Atomscatm 36391  HLchlt 36478  𝑃cpolN 37030  PSubClcpscN 37062
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36081
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  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-rmo 3144  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-iin 4913  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-undef 7931  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-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-pmap 36632  df-polarityN 37031  df-psubclN 37063
This theorem is referenced by:  osumcllem9N  37092
  Copyright terms: Public domain W3C validator