Theorem ltrnatlw 37321

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 1198	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐹‘𝑄) = 𝑄)) → (𝐹‘𝑄) = 𝑄)
2		simpl1 1187	. . . . 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 1247	. . . . 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 1248	. . . . 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 1249	. . . . . 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 487	. . . . . 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 514	. . . . 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 1224	. . . . 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 37320	. . . . 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 1379	. . . 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 415	. . 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 3038	. 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 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   class class class wbr 5068  ‘cfv 6357  lecple 16574  Atomscatm 36401  HLchlt 36488  LHypclh 37122  LTrncltrn 37239

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-p1 17652  df-lat 17658  df-clat 17720  df-oposet 36314  df-ol 36316  df-oml 36317  df-covers 36404  df-ats 36405  df-atl 36436  df-cvlat 36460  df-hlat 36489  df-lhyp 37126  df-laut 37127  df-ldil 37242  df-ltrn 37243  df-trl 37297

This theorem is referenced by:  cdlemg18  37820