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Theorem cdlemf 34672
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 2609 . . 3 (join‘𝐾) = (join‘𝐾)
3 cdlemf.a . . 3 𝐴 = (Atoms‘𝐾)
4 cdlemf.h . . 3 𝐻 = (LHyp‘𝐾)
5 eqid 2609 . . 3 (meet‘𝐾) = (meet‘𝐾)
61, 2, 3, 4, 5cdlemf2 34671 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ∃𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)))
7 simp1l 1077 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
8 simp2l 1079 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑝𝐴)
9 simp3ll 1124 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑝 𝑊)
10 simp2r 1080 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑞𝐴)
11 simp3lr 1125 . . . . . 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 34668 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → ∃𝑓𝑇 (𝑓𝑝) = 𝑞)
147, 8, 9, 10, 11, 13syl122anc 1326 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓𝑇 (𝑓𝑝) = 𝑞)
15 simp3r 1082 . . . . . . . . . . . . 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 6543 . . . . . . . . . . . 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 6542 . . . . . . . . . . 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 1083 . . . . . . . . . . . 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 1081 . . . . . . . . . . . 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 1168 . . . . . . . . . . . 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 1120 . . . . . . . . . . . 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 34271 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇 ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) → (𝑅𝑓) = ((𝑝(join‘𝐾)(𝑓𝑝))(meet‘𝐾)𝑊))
2418, 19, 20, 21, 23syl112anc 1321 . . . . . . . . . . 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 1080 . . . . . . . . . . 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 2653 . . . . . . . . . 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 1255 . . . . . . . . 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 1258 . . . . . . . 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 1248 . . . . . . 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 450 . . . . . 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 2997 . . . . 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 1255 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ((𝑝𝐴𝑞𝐴) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓𝑇 (𝑅𝑓) = 𝑈)))
3433rexlimdvv 3018 . 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 382  w3a 1030   = wceq 1474  wcel 1976  wrex 2896   class class class wbr 4577  cfv 5790  (class class class)co 6527  lecple 15721  joincjn 16713  meetcmee 16714  Atomscatm 33371  HLchlt 33458  LHypclh 34091  LTrncltrn 34208  trLctrl 34266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33060
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-undef 7263  df-map 7723  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-llines 33605  df-lplanes 33606  df-lvols 33607  df-lines 33608  df-psubsp 33610  df-pmap 33611  df-padd 33903  df-lhyp 34095  df-laut 34096  df-ldil 34211  df-ltrn 34212  df-trl 34267
This theorem is referenced by:  cdlemfnid  34673  trlord  34678  dih1dimb2  35351
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