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Theorem cdleme 35325
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 35324 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃𝑓𝑇 (𝑓𝑃) = 𝑄)
6 simp11 1089 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simp2l 1085 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑓𝑇)
8 simp2r 1086 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑧𝑇)
9 simp12 1090 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
10 eqtr3 2642 . . . . . 6 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → (𝑓𝑃) = (𝑧𝑃))
11103ad2ant3 1082 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝑓𝑃) = (𝑧𝑃))
121, 2, 3, 4cdlemd 34971 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇𝑧𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑓𝑃) = (𝑧𝑃)) → 𝑓 = 𝑧)
136, 7, 8, 9, 11, 12syl311anc 1337 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑓 = 𝑧)
14133exp 1261 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑓𝑇𝑧𝑇) → (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧)))
1514ralrimivv 2964 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∀𝑓𝑇𝑧𝑇 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧))
16 fveq1 6147 . . . 4 (𝑓 = 𝑧 → (𝑓𝑃) = (𝑧𝑃))
1716eqeq1d 2623 . . 3 (𝑓 = 𝑧 → ((𝑓𝑃) = 𝑄 ↔ (𝑧𝑃) = 𝑄))
1817reu4 3382 . 2 (∃!𝑓𝑇 (𝑓𝑃) = 𝑄 ↔ (∃𝑓𝑇 (𝑓𝑃) = 𝑄 ∧ ∀𝑓𝑇𝑧𝑇 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧)))
195, 15, 18sylanbrc 697 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909   class class class wbr 4613  cfv 5847  lecple 15869  Atomscatm 34027  HLchlt 34114  LHypclh 34747  LTrncltrn 34864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-riotaBAD 33716
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-undef 7344  df-map 7804  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-p1 16961  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-llines 34261  df-lplanes 34262  df-lvols 34263  df-lines 34264  df-psubsp 34266  df-pmap 34267  df-padd 34559  df-lhyp 34751  df-laut 34752  df-ldil 34867  df-ltrn 34868  df-trl 34923
This theorem is referenced by:  ltrniotaval  35346  cdlemeiota  35350  cdlemksv2  35612  cdlemkuv2  35632
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