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Theorem trlval2 40145
Description: The value of the trace of a lattice translation, given any atom 𝑃 not under the fiducial co-atom 𝑊. Note: this requires only the weaker assumption 𝐾 ∈ Lat; we use 𝐾 ∈ HL for convenience. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlval2.l = (le‘𝐾)
trlval2.j = (join‘𝐾)
trlval2.m = (meet‘𝐾)
trlval2.a 𝐴 = (Atoms‘𝐾)
trlval2.h 𝐻 = (LHyp‘𝐾)
trlval2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlval2.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trlval2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))

Proof of Theorem trlval2
Dummy variables 𝑥 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 39344 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
21anim1i 615 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐾 ∈ Lat ∧ 𝑊𝐻))
3 eqid 2734 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
4 trlval2.l . . . . 5 = (le‘𝐾)
5 trlval2.j . . . . 5 = (join‘𝐾)
6 trlval2.m . . . . 5 = (meet‘𝐾)
7 trlval2.a . . . . 5 𝐴 = (Atoms‘𝐾)
8 trlval2.h . . . . 5 𝐻 = (LHyp‘𝐾)
9 trlval2.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 trlval2.r . . . . 5 𝑅 = ((trL‘𝐾)‘𝑊)
113, 4, 5, 6, 7, 8, 9, 10trlval 40144 . . . 4 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
12113adant3 1131 . . 3 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
13 simp1l 1196 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
14 simp3l 1200 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
153, 7atbase 39270 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1614, 15syl 17 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
173, 8, 9ltrncl 40107 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
1816, 17syld3an3 1408 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
193, 5latjcl 18496 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
2013, 16, 18, 19syl3anc 1370 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
21 simp1r 1197 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
223, 8lhpbase 39980 . . . . . 6 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2321, 22syl 17 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
243, 6latmcl 18497 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) ∈ (Base‘𝐾))
2513, 20, 23, 24syl3anc 1370 . . . 4 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) ∈ (Base‘𝐾))
26 simpl3l 1227 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑃𝐴)
27 simpl3r 1228 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → ¬ 𝑃 𝑊)
28 breq1 5150 . . . . . . . . . 10 (𝑞 = 𝑃 → (𝑞 𝑊𝑃 𝑊))
2928notbid 318 . . . . . . . . 9 (𝑞 = 𝑃 → (¬ 𝑞 𝑊 ↔ ¬ 𝑃 𝑊))
30 id 22 . . . . . . . . . . . 12 (𝑞 = 𝑃𝑞 = 𝑃)
31 fveq2 6906 . . . . . . . . . . . 12 (𝑞 = 𝑃 → (𝐹𝑞) = (𝐹𝑃))
3230, 31oveq12d 7448 . . . . . . . . . . 11 (𝑞 = 𝑃 → (𝑞 (𝐹𝑞)) = (𝑃 (𝐹𝑃)))
3332oveq1d 7445 . . . . . . . . . 10 (𝑞 = 𝑃 → ((𝑞 (𝐹𝑞)) 𝑊) = ((𝑃 (𝐹𝑃)) 𝑊))
3433eqeq2d 2745 . . . . . . . . 9 (𝑞 = 𝑃 → (𝑥 = ((𝑞 (𝐹𝑞)) 𝑊) ↔ 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
3529, 34imbi12d 344 . . . . . . . 8 (𝑞 = 𝑃 → ((¬ 𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ (¬ 𝑃 𝑊𝑥 = ((𝑃 (𝐹𝑃)) 𝑊))))
3635rspcv 3617 . . . . . . 7 (𝑃𝐴 → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) → (¬ 𝑃 𝑊𝑥 = ((𝑃 (𝐹𝑃)) 𝑊))))
3736com23 86 . . . . . 6 (𝑃𝐴 → (¬ 𝑃 𝑊 → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) → 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊))))
3826, 27, 37sylc 65 . . . . 5 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) → 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
39 simp11 1202 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → (𝐾 ∈ Lat ∧ 𝑊𝐻))
40 simp12 1203 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝐹𝑇)
41 simp13l 1287 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝑃𝐴)
42 simp13r 1288 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ¬ 𝑃 𝑊)
43 simp3 1137 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝑞𝐴)
44 simp2 1136 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ¬ 𝑞 𝑊)
454, 5, 6, 7, 8, 9ltrnu 40103 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
4639, 40, 41, 42, 43, 44, 45syl222anc 1385 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
47 eqeq2 2746 . . . . . . . . . . 11 (((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) ↔ 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))
4847biimpd 229 . . . . . . . . . 10 (((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))
4946, 48syl 17 . . . . . . . . 9 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))
50493exp 1118 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (¬ 𝑞 𝑊 → (𝑞𝐴 → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))))
5150com24 95 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → (𝑞𝐴 → (¬ 𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))))
5251ralrimdv 3149 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → ∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
5352adantr 480 . . . . 5 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → ∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
5438, 53impbid 212 . . . 4 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
5525, 54riota5 7416 . . 3 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))) = ((𝑃 (𝐹𝑃)) 𝑊))
5612, 55eqtrd 2774 . 2 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
572, 56syl3an1 1162 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058   class class class wbr 5147  cfv 6562  crio 7386  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  Latclat 18488  Atomscatm 39244  HLchlt 39331  LHypclh 39966  LTrncltrn 40083  trLctrl 40140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-lat 18489  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-lhyp 39970  df-laut 39971  df-ldil 40086  df-ltrn 40087  df-trl 40141
This theorem is referenced by:  trlcl  40146  trlcnv  40147  trljat1  40148  trljat2  40149  trlat  40151  trl0  40152  trlle  40166  trlval3  40169  trlval5  40171  cdlemd6  40185  cdlemf  40545  cdlemg4a  40590  cdlemg4b1  40591  cdlemg4b2  40592  cdlemg4  40599  cdlemg11b  40624  cdlemg13a  40633  cdlemg13  40634  cdlemg17a  40643  cdlemg17dN  40645  cdlemg17e  40647  cdlemg17f  40648  trlcoabs2N  40704  trlcolem  40708  cdlemg42  40711  cdlemg43  40712  cdlemi1  40800  cdlemk4  40816  cdlemk39  40898  dia2dimlem1  41046  dia2dimlem2  41047  dia2dimlem3  41048  cdlemm10N  41100  cdlemn2  41177  cdlemn10  41188  dihjatcclem3  41402
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