Theorem ltrnatlw 35470

Description: If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013.)

Hypotheses

Ref	Expression
ltrn2eq.l	⊢ ≤ = (le‘𝐾)
ltrn2eq.a	⊢ 𝐴 = (Atoms‘𝐾)
ltrn2eq.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrn2eq.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
ltrnatlw	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) → 𝑄 ≤ 𝑊)

Proof of Theorem ltrnatlw

Step	Hyp	Ref	Expression
1		simp3r 1090	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) → (𝐹‘𝑄) = 𝑄)
2		simpl1 1064	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simpl21 1139	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → 𝐹 ∈ 𝑇)
4		simpl22 1140	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5		simpl23 1141	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → 𝑄 ∈ 𝐴)
6		simpr 477	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → ¬ 𝑄 ≤ 𝑊)
7	5, 6	jca 554	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
8		simpl3l 1116	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → (𝐹‘𝑃) ≠ 𝑃)
9		ltrn2eq.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
10		ltrn2eq.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
11		ltrn2eq.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
12		ltrn2eq.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13	9, 10, 11, 12	ltrnatneq 35469	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹‘𝑄) ≠ 𝑄)
14	2, 3, 4, 7, 8, 13	syl131anc 1339	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) ∧ ¬ 𝑄 ≤ 𝑊) → (𝐹‘𝑄) ≠ 𝑄)
15	14	ex 450	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) → (¬ 𝑄 ≤ 𝑊 → (𝐹‘𝑄) ≠ 𝑄))
16	15	necon4bd 2814	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) → ((𝐹‘𝑄) = 𝑄 → 𝑄 ≤ 𝑊))
17	1, 16	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) → 𝑄 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   class class class wbr 4653  ‘cfv 5888  lecple 15948  Atomscatm 34550  HLchlt 34637  LHypclh 35270  LTrncltrn 35387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemg18  35970