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Theorem ltrnmwOLD 35756
 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 35755 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 1081 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp2 1082 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐹𝑇)
3 simp3l 1109 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
4 eqid 2651 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
5 ltrnmwOLD.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atbase 34894 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
73, 6syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
8 simp1r 1106 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
9 ltrnmwOLD.h . . . . . 6 𝐻 = (LHyp‘𝐾)
104, 9lhpbase 35602 . . . . 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 35735 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (𝐹‘(𝑃 𝑊)) = ((𝐹𝑃) (𝐹𝑊)))
151, 2, 7, 11, 14syl112anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹‘(𝑃 𝑊)) = ((𝐹𝑃) (𝐹𝑊)))
16 simp3r 1110 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ¬ 𝑃 𝑊)
17 simp1l 1105 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ HL)
18 hlatl 34965 . . . . . . 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 34922 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑊 ∈ (Base‘𝐾)) → (¬ 𝑃 𝑊 ↔ (𝑃 𝑊) = 0 ))
2319, 3, 11, 22syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (¬ 𝑃 𝑊 ↔ (𝑃 𝑊) = 0 ))
2416, 23mpbid 222 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = 0 )
2524fveq2d 6233 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹‘(𝑃 𝑊)) = (𝐹0 ))
2615, 25eqtr3d 2687 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) (𝐹𝑊)) = (𝐹0 ))
27 hllat 34968 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2817, 27syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
294, 20latref 17100 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑊 ∈ (Base‘𝐾)) → 𝑊 𝑊)
3028, 11, 29syl2anc 694 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 𝑊)
314, 20, 9, 13ltrnval1 35738 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊 𝑊)) → (𝐹𝑊) = 𝑊)
321, 2, 11, 30, 31syl112anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑊) = 𝑊)
3332oveq2d 6706 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) (𝐹𝑊)) = ((𝐹𝑃) 𝑊))
34 hlop 34967 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OP)
3517, 34syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ OP)
364, 21op0cl 34789 . . . 4 (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
3735, 36syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 0 ∈ (Base‘𝐾))
384, 20, 21op0le 34791 . . . 4 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 𝑊)
3935, 11, 38syl2anc 694 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 0 𝑊)
404, 20, 9, 13ltrnval1 35738 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ( 0 ∈ (Base‘𝐾) ∧ 0 𝑊)) → (𝐹0 ) = 0 )
411, 2, 37, 39, 40syl112anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹0 ) = 0 )
4226, 33, 413eqtr3d 2693 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) 𝑊) = 0 )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  meetcmee 16992  0.cp0 17084  Latclat 17092  OPcops 34777  Atomscatm 34868  AtLatcal 34869  HLchlt 34955  LHypclh 35588  LTrncltrn 35705 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709 This theorem is referenced by: (None)
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