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Theorem cdleme 37089
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 37088 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃𝑓𝑇 (𝑓𝑃) = 𝑄)
6 simp11 1183 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simp2l 1179 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑓𝑇)
8 simp2r 1180 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑧𝑇)
9 simp12 1184 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
10 eqtr3 2795 . . . . . 6 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → (𝑓𝑃) = (𝑧𝑃))
11103ad2ant3 1115 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → (𝑓𝑃) = (𝑧𝑃))
121, 2, 3, 4cdlemd 36736 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇𝑧𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑓𝑃) = (𝑧𝑃)) → 𝑓 = 𝑧)
136, 7, 8, 9, 11, 12syl311anc 1364 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓𝑇𝑧𝑇) ∧ ((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄)) → 𝑓 = 𝑧)
14133exp 1099 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑓𝑇𝑧𝑇) → (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧)))
1514ralrimivv 3134 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∀𝑓𝑇𝑧𝑇 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧))
16 fveq1 6492 . . . 4 (𝑓 = 𝑧 → (𝑓𝑃) = (𝑧𝑃))
1716eqeq1d 2774 . . 3 (𝑓 = 𝑧 → ((𝑓𝑃) = 𝑄 ↔ (𝑧𝑃) = 𝑄))
1817reu4 3630 . 2 (∃!𝑓𝑇 (𝑓𝑃) = 𝑄 ↔ (∃𝑓𝑇 (𝑓𝑃) = 𝑄 ∧ ∀𝑓𝑇𝑧𝑇 (((𝑓𝑃) = 𝑄 ∧ (𝑧𝑃) = 𝑄) → 𝑓 = 𝑧)))
195, 15, 18sylanbrc 575 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 387  w3a 1068   = wceq 1507  wcel 2048  wral 3082  wrex 3083  ∃!wreu 3084   class class class wbr 4923  cfv 6182  lecple 16418  Atomscatm 35792  HLchlt 35879  LHypclh 36513  LTrncltrn 36630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-riotaBAD 35482
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-undef 7735  df-map 8200  df-proset 17386  df-poset 17404  df-plt 17416  df-lub 17432  df-glb 17433  df-join 17434  df-meet 17435  df-p0 17497  df-p1 17498  df-lat 17504  df-clat 17566  df-oposet 35705  df-ol 35707  df-oml 35708  df-covers 35795  df-ats 35796  df-atl 35827  df-cvlat 35851  df-hlat 35880  df-llines 36027  df-lplanes 36028  df-lvols 36029  df-lines 36030  df-psubsp 36032  df-pmap 36033  df-padd 36325  df-lhyp 36517  df-laut 36518  df-ldil 36633  df-ltrn 36634  df-trl 36688
This theorem is referenced by:  ltrniotaval  37110  cdlemeiota  37114  cdlemksv2  37376  cdlemkuv2  37396
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