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Theorem trlval2 36326
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 35526 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
21anim1i 608 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐾 ∈ Lat ∧ 𝑊𝐻))
3 eqid 2778 . . . . 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 36325 . . . 4 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
12113adant3 1123 . . 3 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
13 simp1l 1211 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
14 simp3l 1215 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
153, 7atbase 35452 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1614, 15syl 17 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
173, 8, 9ltrncl 36288 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
1816, 17syld3an3 1477 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
193, 5latjcl 17448 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
2013, 16, 18, 19syl3anc 1439 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
21 simp1r 1212 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
223, 8lhpbase 36161 . . . . . 6 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2321, 22syl 17 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
243, 6latmcl 17449 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) ∈ (Base‘𝐾))
2513, 20, 23, 24syl3anc 1439 . . . 4 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) ∈ (Base‘𝐾))
26 simpl3l 1258 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑃𝐴)
27 simpl3r 1260 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → ¬ 𝑃 𝑊)
28 breq1 4891 . . . . . . . . . 10 (𝑞 = 𝑃 → (𝑞 𝑊𝑃 𝑊))
2928notbid 310 . . . . . . . . 9 (𝑞 = 𝑃 → (¬ 𝑞 𝑊 ↔ ¬ 𝑃 𝑊))
30 id 22 . . . . . . . . . . . 12 (𝑞 = 𝑃𝑞 = 𝑃)
31 fveq2 6448 . . . . . . . . . . . 12 (𝑞 = 𝑃 → (𝐹𝑞) = (𝐹𝑃))
3230, 31oveq12d 6942 . . . . . . . . . . 11 (𝑞 = 𝑃 → (𝑞 (𝐹𝑞)) = (𝑃 (𝐹𝑃)))
3332oveq1d 6939 . . . . . . . . . 10 (𝑞 = 𝑃 → ((𝑞 (𝐹𝑞)) 𝑊) = ((𝑃 (𝐹𝑃)) 𝑊))
3433eqeq2d 2788 . . . . . . . . 9 (𝑞 = 𝑃 → (𝑥 = ((𝑞 (𝐹𝑞)) 𝑊) ↔ 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
3529, 34imbi12d 336 . . . . . . . 8 (𝑞 = 𝑃 → ((¬ 𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ (¬ 𝑃 𝑊𝑥 = ((𝑃 (𝐹𝑃)) 𝑊))))
3635rspcv 3507 . . . . . . 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 1217 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → (𝐾 ∈ Lat ∧ 𝑊𝐻))
40 simp12 1218 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝐹𝑇)
41 simp13l 1344 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝑃𝐴)
42 simp13r 1345 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ¬ 𝑃 𝑊)
43 simp3 1129 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝑞𝐴)
44 simp2 1128 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ¬ 𝑞 𝑊)
454, 5, 6, 7, 8, 9ltrnu 36284 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
4639, 40, 41, 42, 43, 44, 45syl222anc 1454 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
47 eqeq2 2789 . . . . . . . . . . 11 (((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) ↔ 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))
4847biimpd 221 . . . . . . . . . 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 1109 . . . . . . . 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 3150 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → ∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
5352adantr 474 . . . . 5 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → ∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
5438, 53impbid 204 . . . 4 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
5525, 54riota5 6911 . . 3 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))) = ((𝑃 (𝐹𝑃)) 𝑊))
5612, 55eqtrd 2814 . 2 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
572, 56syl3an1 1163 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 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090   class class class wbr 4888  cfv 6137  crio 6884  (class class class)co 6924  Basecbs 16266  lecple 16356  joincjn 17341  meetcmee 17342  Latclat 17442  Atomscatm 35426  HLchlt 35513  LHypclh 36147  LTrncltrn 36264  trLctrl 36321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-lub 17371  df-glb 17372  df-join 17373  df-meet 17374  df-lat 17443  df-ats 35430  df-atl 35461  df-cvlat 35485  df-hlat 35514  df-lhyp 36151  df-laut 36152  df-ldil 36267  df-ltrn 36268  df-trl 36322
This theorem is referenced by:  trlcl  36327  trlcnv  36328  trljat1  36329  trljat2  36330  trlat  36332  trl0  36333  trlle  36347  trlval3  36350  trlval5  36352  cdlemd6  36366  cdlemf  36726  cdlemg4a  36771  cdlemg4b1  36772  cdlemg4b2  36773  cdlemg4  36780  cdlemg11b  36805  cdlemg13a  36814  cdlemg13  36815  cdlemg17a  36824  cdlemg17dN  36826  cdlemg17e  36828  cdlemg17f  36829  trlcoabs2N  36885  trlcolem  36889  cdlemg42  36892  cdlemg43  36893  cdlemi1  36981  cdlemk4  36997  cdlemk39  37079  dia2dimlem1  37227  dia2dimlem2  37228  dia2dimlem3  37229  cdlemm10N  37281  cdlemn2  37358  cdlemn10  37369  dihjatcclem3  37583
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