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Theorem ltrnmwOLD 34254
Description: Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.) Obsolete version of ltrnmw 34253 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ltrnmwOLD.l = (le‘𝐾)
ltrnmwOLD.m = (meet‘𝐾)
ltrnmwOLD.z 0 = (0.‘𝐾)
ltrnmwOLD.a 𝐴 = (Atoms‘𝐾)
ltrnmwOLD.h 𝐻 = (LHyp‘𝐾)
ltrnmwOLD.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrnmwOLD (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) 𝑊) = 0 )

Proof of Theorem ltrnmwOLD
StepHypRef Expression
1 simp1 1053 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp2 1054 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐹𝑇)
3 simp3l 1081 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
4 eqid 2604 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
5 ltrnmwOLD.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atbase 33392 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
73, 6syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
8 simp1r 1078 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
9 ltrnmwOLD.h . . . . . 6 𝐻 = (LHyp‘𝐾)
104, 9lhpbase 34100 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
118, 10syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
12 ltrnmwOLD.m . . . . 5 = (meet‘𝐾)
13 ltrnmwOLD.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
144, 12, 9, 13ltrnm 34233 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (𝐹‘(𝑃 𝑊)) = ((𝐹𝑃) (𝐹𝑊)))
151, 2, 7, 11, 14syl112anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹‘(𝑃 𝑊)) = ((𝐹𝑃) (𝐹𝑊)))
16 simp3r 1082 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ¬ 𝑃 𝑊)
17 simp1l 1077 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ HL)
18 hlatl 33463 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1917, 18syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ AtLat)
20 ltrnmwOLD.l . . . . . . 7 = (le‘𝐾)
21 ltrnmwOLD.z . . . . . . 7 0 = (0.‘𝐾)
224, 20, 12, 21, 5atnle 33420 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑊 ∈ (Base‘𝐾)) → (¬ 𝑃 𝑊 ↔ (𝑃 𝑊) = 0 ))
2319, 3, 11, 22syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (¬ 𝑃 𝑊 ↔ (𝑃 𝑊) = 0 ))
2416, 23mpbid 220 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = 0 )
2524fveq2d 6087 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹‘(𝑃 𝑊)) = (𝐹0 ))
2615, 25eqtr3d 2640 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) (𝐹𝑊)) = (𝐹0 ))
27 hllat 33466 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2817, 27syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
294, 20latref 16817 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑊 ∈ (Base‘𝐾)) → 𝑊 𝑊)
3028, 11, 29syl2anc 690 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 𝑊)
314, 20, 9, 13ltrnval1 34236 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊 𝑊)) → (𝐹𝑊) = 𝑊)
321, 2, 11, 30, 31syl112anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑊) = 𝑊)
3332oveq2d 6538 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) (𝐹𝑊)) = ((𝐹𝑃) 𝑊))
34 hlop 33465 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OP)
3517, 34syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ OP)
364, 21op0cl 33287 . . . 4 (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
3735, 36syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 0 ∈ (Base‘𝐾))
384, 20, 21op0le 33289 . . . 4 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 𝑊)
3935, 11, 38syl2anc 690 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 0 𝑊)
404, 20, 9, 13ltrnval1 34236 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ( 0 ∈ (Base‘𝐾) ∧ 0 𝑊)) → (𝐹0 ) = 0 )
411, 2, 37, 39, 40syl112anc 1321 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹0 ) = 0 )
4226, 33, 413eqtr3d 2646 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) 𝑊) = 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  lecple 15716  meetcmee 16709  0.cp0 16801  Latclat 16809  OPcops 33275  Atomscatm 33366  AtLatcal 33367  HLchlt 33453  LHypclh 34086  LTrncltrn 34203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-map 7718  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-lat 16810  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-lhyp 34090  df-laut 34091  df-ldil 34206  df-ltrn 34207
This theorem is referenced by: (None)
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