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Theorem cdlemf 36168
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 2651 . . 3 (join‘𝐾) = (join‘𝐾)
3 cdlemf.a . . 3 𝐴 = (Atoms‘𝐾)
4 cdlemf.h . . 3 𝐻 = (LHyp‘𝐾)
5 eqid 2651 . . 3 (meet‘𝐾) = (meet‘𝐾)
61, 2, 3, 4, 5cdlemf2 36167 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ∃𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)))
7 simp1l 1105 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
8 simp2l 1107 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑝𝐴)
9 simp3ll 1152 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑝 𝑊)
10 simp2r 1108 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑞𝐴)
11 simp3lr 1153 . . . . . 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 36164 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → ∃𝑓𝑇 (𝑓𝑝) = 𝑞)
147, 8, 9, 10, 11, 13syl122anc 1375 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) ∧ (𝑝𝐴𝑞𝐴) ∧ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓𝑇 (𝑓𝑝) = 𝑞)
15 simp3r 1110 . . . . . . . . . . . . 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 6706 . . . . . . . . . . . 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 6705 . . . . . . . . . . 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 1111 . . . . . . . . . . . 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 1109 . . . . . . . . . . . 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 1196 . . . . . . . . . . . 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 1148 . . . . . . . . . . . 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 35768 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇 ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) → (𝑅𝑓) = ((𝑝(join‘𝐾)(𝑓𝑝))(meet‘𝐾)𝑊))
2418, 19, 20, 21, 23syl112anc 1370 . . . . . . . . . . 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 1108 . . . . . . . . . . 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 2695 . . . . . . . . . 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 1283 . . . . . . . . 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 1286 . . . . . . . 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 1275 . . . . . . 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 451 . . . . . 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 3044 . . . . 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 1283 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐴𝑈 𝑊)) → ((𝑝𝐴𝑞𝐴) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓𝑇 (𝑅𝑓) = 𝑈)))
3433rexlimdvv 3066 . 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 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  trLctrl 35763
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  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  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-iin 4555  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-1st 7210  df-2nd 7211  df-undef 7444  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-p1 17087  df-lat 17093  df-clat 17155  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-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764
This theorem is referenced by:  cdlemfnid  36169  trlord  36174  dih1dimb2  36847
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