Theorem cdleme 40554

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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 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.

Step	Hyp	Ref	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)
5	1, 2, 3, 4	cdleme50ex 40553	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)
6		simp11 1204	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		simp2l 1200	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → 𝑓 ∈ 𝑇)
8		simp2r 1201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → 𝑧 ∈ 𝑇)
9		simp12 1205	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
10		eqtr3 2751	. . . . . 6 ⊢ (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → (𝑓‘𝑃) = (𝑧‘𝑃))
11	10	3ad2ant3 1135	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → (𝑓‘𝑃) = (𝑧‘𝑃))
12	1, 2, 3, 4	cdlemd 40201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑓‘𝑃) = (𝑧‘𝑃)) → 𝑓 = 𝑧)
13	6, 7, 8, 9, 11, 12	syl311anc 1386	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → 𝑓 = 𝑧)
14	13	3exp 1119	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → 𝑓 = 𝑧)))
15	14	ralrimivv 3178	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∀𝑓 ∈ 𝑇 ∀𝑧 ∈ 𝑇 (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → 𝑓 = 𝑧))
16		fveq1 6857	. . . 4 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑃) = (𝑧‘𝑃))
17	16	eqeq1d 2731	. . 3 ⊢ (𝑓 = 𝑧 → ((𝑓‘𝑃) = 𝑄 ↔ (𝑧‘𝑃) = 𝑄))
18	17	reu4 3702	. 2 ⊢ (∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄 ↔ (∃𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄 ∧ ∀𝑓 ∈ 𝑇 ∀𝑧 ∈ 𝑇 (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → 𝑓 = 𝑧)))
19	5, 15, 18	sylanbrc 583	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ∃!wreu 3352   class class class wbr 5107  ‘cfv 6511  lecple 17227  Atomscatm 39256  HLchlt 39343  LHypclh 39978  LTrncltrn 40095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-riotaBAD 38946

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-undef 8252  df-map 8801  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-llines 39492  df-lplanes 39493  df-lvols 39494  df-lines 39495  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982  df-laut 39983  df-ldil 40098  df-ltrn 40099  df-trl 40153

This theorem is referenced by:  ltrniotaval  40575  cdlemeiota  40579  cdlemksv2  40841  cdlemkuv2  40861