Theorem cdleme 41883

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 41882	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)
6		simp11 1222	. . . . 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 1218	. . . . 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 1219	. . . . 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 1223	. . . . 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 2785	. . . . . 6 ⊢ (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → (𝑓‘𝑃) = (𝑧‘𝑃))
11	10	3ad2ant3 1153	. . . . 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 41530	. . . . 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 1412	. . . 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 1137	. . 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 3207	. 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 6915	. . . 4 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑃) = (𝑧‘𝑃))
17	16	eqeq1d 2765	. . 3 ⊢ (𝑓 = 𝑧 → ((𝑓‘𝑃) = 𝑄 ↔ (𝑧‘𝑃) = 𝑄))
18	17	reu4 3692	. 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 595	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367   class class class wbr 5108  ‘cfv 6566  lecple 17479  Atomscatm 40586  HLchlt 40673  LHypclh 41307  LTrncltrn 41424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-riotaBAD 40276

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-undef 8328  df-map 8887  df-proset 18518  df-poset 18537  df-plt 18552  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-p0 18647  df-p1 18648  df-lat 18656  df-clat 18723  df-oposet 40499  df-ol 40501  df-oml 40502  df-covers 40589  df-ats 40590  df-atl 40621  df-cvlat 40645  df-hlat 40674  df-llines 40821  df-lplanes 40822  df-lvols 40823  df-lines 40824  df-psubsp 40826  df-pmap 40827  df-padd 41119  df-lhyp 41311  df-laut 41312  df-ldil 41427  df-ltrn 41428  df-trl 41482

This theorem is used by:  ltrniotaval  41904  cdlemeiota  41908  cdlemksv2  42170  cdlemkuv2  42190