Theorem cdleme 40599

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 40598	. 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 2753	. . . . . 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 40246	. . . . 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 3173	. 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 6816	. . . 4 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑃) = (𝑧‘𝑃))
17	16	eqeq1d 2733	. . 3 ⊢ (𝑓 = 𝑧 → ((𝑓‘𝑃) = 𝑄 ↔ (𝑧‘𝑃) = 𝑄))
18	17	reu4 3685	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ∃!wreu 3344   class class class wbr 5086  ‘cfv 6476  lecple 17163  Atomscatm 39302  HLchlt 39389  LHypclh 40023  LTrncltrn 40140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-riotaBAD 38992

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-undef 8198  df-map 8747  df-proset 18195  df-poset 18214  df-plt 18229  df-lub 18245  df-glb 18246  df-join 18247  df-meet 18248  df-p0 18324  df-p1 18325  df-lat 18333  df-clat 18400  df-oposet 39215  df-ol 39217  df-oml 39218  df-covers 39305  df-ats 39306  df-atl 39337  df-cvlat 39361  df-hlat 39390  df-llines 39537  df-lplanes 39538  df-lvols 39539  df-lines 39540  df-psubsp 39542  df-pmap 39543  df-padd 39835  df-lhyp 40027  df-laut 40028  df-ldil 40143  df-ltrn 40144  df-trl 40198

This theorem is referenced by:  ltrniotaval  40620  cdlemeiota  40624  cdlemksv2  40886  cdlemkuv2  40906