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Theorem cdlemf 40068
Description: Lemma F in [Crawley] p. 116. If u is an atom under w, there exists a translation whose trace is u. (Contributed by NM, 12-Apr-2013.)
Hypotheses
Ref Expression
cdlemf.l = (le‘𝐾)
cdlemf.a 𝐴 = (Atoms‘𝐾)
cdlemf.h 𝐻 = (LHyp‘𝐾)
cdlemf.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemf.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemf (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ∃𝑓𝑇 (𝑅𝑓) = 𝑈)
Distinct variable groups:   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   ,𝑓   𝑇,𝑓   𝑈,𝑓   𝑓,𝑊
Allowed substitution hint:   𝑅(𝑓)

Proof of Theorem cdlemf
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemf.l . . 3 = (le‘𝐾)
2 eqid 2728 . . 3 (join‘𝐾) = (join‘𝐾)
3 cdlemf.a . . 3 𝐴 = (Atoms‘𝐾)
4 cdlemf.h . . 3 𝐻 = (LHyp‘𝐾)
5 eqid 2728 . . 3 (meet‘𝐾) = (meet‘𝐾)
61, 2, 3, 4, 5cdlemf2 40067 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ∃𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)))
7 simp1l 1194 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
8 simp2l 1196 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑝𝐴)
9 simp3ll 1241 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑝 𝑊)
10 simp2r 1197 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑞𝐴)
11 simp3lr 1242 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑞 𝑊)
12 cdlemf.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
131, 3, 4, 12cdleme50ex 40064 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → ∃𝑓𝑇 (𝑓𝑝) = 𝑞)
147, 8, 9, 10, 11, 13syl122anc 1376 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓𝑇 (𝑓𝑝) = 𝑞)
15 simp3r 1199 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → (𝑓𝑝) = 𝑞)
1615oveq2d 7442 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → (𝑝(join‘𝐾)(𝑓𝑝)) = (𝑝(join‘𝐾)𝑞))
1716oveq1d 7441 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → ((𝑝(join‘𝐾)(𝑓𝑝))(meet‘𝐾)𝑊) = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))
18 simp11 1200 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
19 simp3l 1198 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → 𝑓𝑇)
20 simp13l 1285 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → 𝑝𝐴)
21 simp2ll 1237 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → ¬ 𝑝 𝑊)
22 cdlemf.r . . . . . . . . . . . . 13 𝑅 = ((trL‘𝐾)‘𝑊)
231, 2, 5, 3, 4, 12, 22trlval2 39668 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇 ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) → (𝑅𝑓) = ((𝑝(join‘𝐾)(𝑓𝑝))(meet‘𝐾)𝑊))
2418, 19, 20, 21, 23syl112anc 1371 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → (𝑅𝑓) = ((𝑝(join‘𝐾)(𝑓𝑝))(meet‘𝐾)𝑊))
25 simp2r 1197 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))
2617, 24, 253eqtr4d 2778 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓𝑇 ∧ (𝑓𝑝) = 𝑞)) → (𝑅𝑓) = 𝑈)
27263exp 1116 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑝𝐴𝑞𝐴)) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ((𝑓𝑇 ∧ (𝑓𝑝) = 𝑞) → (𝑅𝑓) = 𝑈)))
28273expia 1118 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ((𝑝𝐴𝑞𝐴) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ((𝑓𝑇 ∧ (𝑓𝑝) = 𝑞) → (𝑅𝑓) = 𝑈))))
29283imp 1108 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ((𝑓𝑇 ∧ (𝑓𝑝) = 𝑞) → (𝑅𝑓) = 𝑈))
3029expd 414 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝑓𝑇 → ((𝑓𝑝) = 𝑞 → (𝑅𝑓) = 𝑈)))
3130reximdvai 3162 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (∃𝑓𝑇 (𝑓𝑝) = 𝑞 → ∃𝑓𝑇 (𝑅𝑓) = 𝑈))
3214, 31mpd 15 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓𝑇 (𝑅𝑓) = 𝑈)
33323exp 1116 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ((𝑝𝐴𝑞𝐴) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓𝑇 (𝑅𝑓) = 𝑈)))
3433rexlimdvv 3208 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → (∃𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓𝑇 (𝑅𝑓) = 𝑈))
356, 34mpd 15 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ∃𝑓𝑇 (𝑅𝑓) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3067   class class class wbr 5152  cfv 6553  (class class class)co 7426  lecple 17247  joincjn 18310  meetcmee 18311  Atomscatm 38767  HLchlt 38854  LHypclh 39489  LTrncltrn 39606  trLctrl 39663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-riotaBAD 38457
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-undef 8285  df-map 8853  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664
This theorem is referenced by:  cdlemfnid  40069  trlord  40074  dih1dimb2  40746
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