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Theorem dihmeetlem17N 37101
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem14.b 𝐵 = (Base‘𝐾)
dihmeetlem14.l = (le‘𝐾)
dihmeetlem14.h 𝐻 = (LHyp‘𝐾)
dihmeetlem14.j = (join‘𝐾)
dihmeetlem14.m = (meet‘𝐾)
dihmeetlem14.a 𝐴 = (Atoms‘𝐾)
dihmeetlem14.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem14.s = (LSSum‘𝑈)
dihmeetlem14.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem17.o 0 = (0.‘𝐾)
Assertion
Ref Expression
dihmeetlem17N ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑌 𝑝) = 0 )

Proof of Theorem dihmeetlem17N
StepHypRef Expression
1 simpl1l 1286 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → 𝐾 ∈ HL)
21hllatd 35142 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → 𝐾 ∈ Lat)
3 simpl3l 1294 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → 𝑝𝐴)
4 dihmeetlem14.b . . . . 5 𝐵 = (Base‘𝐾)
5 dihmeetlem14.a . . . . 5 𝐴 = (Atoms‘𝐾)
64, 5atbase 35067 . . . 4 (𝑝𝐴𝑝𝐵)
73, 6syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → 𝑝𝐵)
8 simpr1 1241 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → 𝑌𝐵)
9 dihmeetlem14.m . . . 4 = (meet‘𝐾)
104, 9latmcom 17276 . . 3 ((𝐾 ∈ Lat ∧ 𝑝𝐵𝑌𝐵) → (𝑝 𝑌) = (𝑌 𝑝))
112, 7, 8, 10syl3anc 1483 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑝 𝑌) = (𝑌 𝑝))
12 simpl1 1235 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
13 simpl2 1237 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
14 simpl3 1239 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑝𝐴 ∧ ¬ 𝑝 𝑊))
15 simpr2 1243 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑋 𝑌) 𝑊)
16 simpr3 1245 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → 𝑝 𝑋)
17 dihmeetlem14.l . . . . 5 = (le‘𝐾)
18 dihmeetlem14.j . . . . 5 = (join‘𝐾)
19 dihmeetlem14.h . . . . 5 𝐻 = (LHyp‘𝐾)
204, 17, 18, 9, 5, 19lhpmcvr4N 35804 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → ¬ 𝑝 𝑌)
2112, 13, 14, 8, 15, 16, 20syl123anc 1499 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → ¬ 𝑝 𝑌)
22 hlatl 35138 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
231, 22syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → 𝐾 ∈ AtLat)
24 dihmeetlem17.o . . . . 5 0 = (0.‘𝐾)
254, 17, 9, 24, 5atnle 35095 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑝𝐴𝑌𝐵) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = 0 ))
2623, 3, 8, 25syl3anc 1483 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (¬ 𝑝 𝑌 ↔ (𝑝 𝑌) = 0 ))
2721, 26mpbid 223 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑝 𝑌) = 0 )
2811, 27eqtr3d 2842 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑌 𝑝) = 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156   class class class wbr 4844  cfv 6097  (class class class)co 6870  Basecbs 16064  lecple 16156  joincjn 17145  meetcmee 17146  0.cp0 17238  Latclat 17246  LSSumclsm 18246  Atomscatm 35041  AtLatcal 35042  HLchlt 35128  LHypclh 35762  DVecHcdvh 36856  DIsoHcdih 37006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-proset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-covers 35044  df-ats 35045  df-atl 35076  df-cvlat 35100  df-hlat 35129  df-lhyp 35766
This theorem is referenced by:  dihmeetlem18N  37102
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