Theorem colinearperm4 36029

Description: Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Assertion

Ref	Expression
colinearperm4	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐶 Colinear ⟨𝐴, 𝐵⟩))

Proof of Theorem colinearperm4

Step	Hyp	Ref	Expression
1		colinearperm3 36027	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐵 Colinear ⟨𝐶, 𝐴⟩))
2		3anrot 1100	. . 3 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)))
3		colinearperm3 36027	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → (𝐵 Colinear ⟨𝐶, 𝐴⟩ ↔ 𝐶 Colinear ⟨𝐴, 𝐵⟩))
4	2, 3	sylan2b 593	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Colinear ⟨𝐶, 𝐴⟩ ↔ 𝐶 Colinear ⟨𝐴, 𝐵⟩))
5	1, 4	bitrd 279	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐶 Colinear ⟨𝐴, 𝐵⟩))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166  ‘cfv 6573  ℕcn 12293  𝔼cee 28921   Colinear ccolin 36001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-iota 6525  df-fv 6581  df-oprab 7452  df-colinear 36003

This theorem is referenced by:  colinearperm5  36030  btwncolinear5  36037