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Theorem cdleme 40543
Description: Lemma E in [Crawley] p. 113. If p,q are atoms not under hyperplane w, then there is a unique translation f such that f(p) = q. (Contributed by NM, 11-Apr-2013.)
Hypotheses
Ref Expression
cdleme.l = (le‘𝐾)
cdleme.a 𝐴 = (Atoms‘𝐾)
cdleme.h 𝐻 = (LHyp‘𝐾)
cdleme.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdleme (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃!𝑓𝑇 (𝑓𝑃) = 𝑄)
Distinct variable groups:   𝐴,𝑓   𝑓,𝐾   ,𝑓   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,𝑊   𝑓,𝐻

Proof of Theorem cdleme
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cdleme.l . . 3 = (le‘𝐾)
2 cdleme.a . . 3 𝐴 = (Atoms‘𝐾)
3 cdleme.h . . 3 𝐻 = (LHyp‘𝐾)
4 cdleme.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
51, 2, 3, 4cdleme50ex 40542 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃𝑓𝑇 (𝑓𝑃) = 𝑄)
6 simp11 1202 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simp2l 1198 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑓𝑇)
8 simp2r 1199 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑧𝑇)
9 simp12 1203 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
10 eqtr3 2761 . . . . . 6 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → (𝑓𝑃) = (𝑧𝑃))
11103ad2ant3 1134 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝑓𝑃) = (𝑧𝑃))
121, 2, 3, 4cdlemd 40190 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇𝑧𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑓𝑃) = (𝑧𝑃)) → 𝑓 = 𝑧)
136, 7, 8, 9, 11, 12syl311anc 1383 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑓 = 𝑧)
14133exp 1118 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑓𝑇𝑧𝑇) → (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧)))
1514ralrimivv 3198 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∀𝑓𝑇𝑧𝑇 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧))
16 fveq1 6906 . . . 4 (𝑓 = 𝑧 → (𝑓𝑃) = (𝑧𝑃))
1716eqeq1d 2737 . . 3 (𝑓 = 𝑧 → ((𝑓𝑃) = 𝑄 ↔ (𝑧𝑃) = 𝑄))
1817reu4 3740 . 2 (∃!𝑓𝑇 (𝑓𝑃) = 𝑄 ↔ (∃𝑓𝑇 (𝑓𝑃) = 𝑄 ∧ ∀𝑓𝑇𝑧𝑇 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧)))
195, 15, 18sylanbrc 583 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃!𝑓𝑇 (𝑓𝑃) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  ∃!wreu 3376   class class class wbr 5148  cfv 6563  lecple 17305  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-undef 8297  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142
This theorem is referenced by:  ltrniotaval  40564  cdlemeiota  40568  cdlemksv2  40830  cdlemkuv2  40850
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