Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme4a Structured version   Visualization version   GIF version

Theorem cdleme4a 37242
Description: Part of proof of Lemma E in [Crawley] p. 114 top. 𝐺 represents fs(r). Auxiliary lemma derived from cdleme5 37243. We show fs(r) p q. (Contributed by NM, 10-Nov-2012.)
Hypotheses
Ref Expression
cdleme4.l = (le‘𝐾)
cdleme4.j = (join‘𝐾)
cdleme4.m = (meet‘𝐾)
cdleme4.a 𝐴 = (Atoms‘𝐾)
cdleme4.h 𝐻 = (LHyp‘𝐾)
cdleme4.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme4.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme4.g 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme4a (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝐺 (𝑃 𝑄))

Proof of Theorem cdleme4a
StepHypRef Expression
1 cdleme4.g . 2 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
2 simp1l 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝐾 ∈ HL)
32hllatd 36367 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝐾 ∈ Lat)
4 simp21 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝑃𝐴)
5 simp22 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝑄𝐴)
6 eqid 2826 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
7 cdleme4.j . . . . 5 = (join‘𝐾)
8 cdleme4.a . . . . 5 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 36370 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
102, 4, 5, 9syl3anc 1365 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp1r 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝑊𝐻)
12 simp3 1132 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝑆𝐴)
13 cdleme4.l . . . . . 6 = (le‘𝐾)
14 cdleme4.m . . . . . 6 = (meet‘𝐾)
15 cdleme4.h . . . . . 6 𝐻 = (LHyp‘𝐾)
16 cdleme4.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
17 cdleme4.f . . . . . 6 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
1813, 7, 14, 8, 15, 16, 17, 6cdleme1b 37229 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → 𝐹 ∈ (Base‘𝐾))
192, 11, 4, 5, 12, 18syl23anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝐹 ∈ (Base‘𝐾))
20 simp23 1202 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝑅𝐴)
216, 7, 8hlatjcl 36370 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
222, 20, 12, 21syl3anc 1365 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
236, 15lhpbase 37001 . . . . . 6 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2411, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝑊 ∈ (Base‘𝐾))
256, 14latmcl 17652 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑅 𝑆) 𝑊) ∈ (Base‘𝐾))
263, 22, 24, 25syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → ((𝑅 𝑆) 𝑊) ∈ (Base‘𝐾))
276, 7latjcl 17651 . . . 4 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑅 𝑆) 𝑊) ∈ (Base‘𝐾)) → (𝐹 ((𝑅 𝑆) 𝑊)) ∈ (Base‘𝐾))
283, 19, 26, 27syl3anc 1365 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → (𝐹 ((𝑅 𝑆) 𝑊)) ∈ (Base‘𝐾))
296, 13, 14latmle1 17676 . . 3 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ((𝑅 𝑆) 𝑊)) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) (𝑃 𝑄))
303, 10, 28, 29syl3anc 1365 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) (𝑃 𝑄))
311, 30eqbrtrid 5098 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑆𝐴) → 𝐺 (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5063  cfv 6352  (class class class)co 7148  Basecbs 16473  lecple 16562  joincjn 17544  meetcmee 17545  Latclat 17645  Atomscatm 36266  HLchlt 36353  LHypclh 36987
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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-lat 17646  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-lhyp 36991
This theorem is referenced by:  cdleme18c  37296  cdleme41sn3a  37436
  Copyright terms: Public domain W3C validator