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Theorem dihmeetlem17N 40189
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem14.b 𝐡 = (Baseβ€˜πΎ)
dihmeetlem14.l ≀ = (leβ€˜πΎ)
dihmeetlem14.h 𝐻 = (LHypβ€˜πΎ)
dihmeetlem14.j ∨ = (joinβ€˜πΎ)
dihmeetlem14.m ∧ = (meetβ€˜πΎ)
dihmeetlem14.a 𝐴 = (Atomsβ€˜πΎ)
dihmeetlem14.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dihmeetlem14.s βŠ• = (LSSumβ€˜π‘ˆ)
dihmeetlem14.i 𝐼 = ((DIsoHβ€˜πΎ)β€˜π‘Š)
dihmeetlem17.o 0 = (0.β€˜πΎ)
Assertion
Ref Expression
dihmeetlem17N ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (π‘Œ ∧ 𝑝) = 0 )
Proof of Theorem dihmeetlem17N
StepHypRef Expression
1 simpl1l 1224 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
21hllatd 38229 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
3 simpl3l 1228 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ 𝑝 ∈ 𝐴)
4 dihmeetlem14.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
5 dihmeetlem14.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38154 . . . 4 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ 𝐡)
73, 6syl 17 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ 𝑝 ∈ 𝐡)
8 simpr1 1194 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ π‘Œ ∈ 𝐡)
9 dihmeetlem14.m . . . 4 ∧ = (meetβ€˜πΎ)
104, 9latmcom 18415 . . 3 ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑝 ∧ π‘Œ) = (π‘Œ ∧ 𝑝))
112, 7, 8, 10syl3anc 1371 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (𝑝 ∧ π‘Œ) = (π‘Œ ∧ 𝑝))
12 simpl1 1191 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
13 simpl2 1192 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š))
14 simpl3 1193 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š))
15 simpr2 1195 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (𝑋 ∧ π‘Œ) ≀ π‘Š)
16 simpr3 1196 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ 𝑝 ≀ 𝑋)
17 dihmeetlem14.l . . . . 5 ≀ = (leβ€˜πΎ)
18 dihmeetlem14.j . . . . 5 ∨ = (joinβ€˜πΎ)
19 dihmeetlem14.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
204, 17, 18, 9, 5, 19lhpmcvr4N 38892 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ Β¬ 𝑝 ≀ π‘Œ)
2112, 13, 14, 8, 15, 16, 20syl123anc 1387 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ Β¬ 𝑝 ≀ π‘Œ)
22 hlatl 38225 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
231, 22syl 17 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ 𝐾 ∈ AtLat)
24 dihmeetlem17.o . . . . 5 0 = (0.β€˜πΎ)
254, 17, 9, 24, 5atnle 38182 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑝 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡) β†’ (Β¬ 𝑝 ≀ π‘Œ ↔ (𝑝 ∧ π‘Œ) = 0 ))
2623, 3, 8, 25syl3anc 1371 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (Β¬ 𝑝 ≀ π‘Œ ↔ (𝑝 ∧ π‘Œ) = 0 ))
2721, 26mpbid 231 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (𝑝 ∧ π‘Œ) = 0 )
2811, 27eqtr3d 2774 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Š ∧ 𝑝 ≀ 𝑋)) β†’ (π‘Œ ∧ 𝑝) = 0 )
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  lecple 17203  joincjn 18263  meetcmee 18264  0.cp0 18375  Latclat 18383  LSSumclsm 19501  Atomscatm 38128  AtLatcal 38129  HLchlt 38215  LHypclh 38850  DVecHcdvh 39944  DIsoHcdih 40094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-lat 18384  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-lhyp 38854
This theorem is referenced by:  dihmeetlem18N  40190
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