Theorem segleantisym 37058

Description: Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)

Assertion

Ref	Expression
segleantisym	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩))

Proof of Theorem segleantisym

Dummy variables 𝑦 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsegle 37051	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
2		brsegle2 37052	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩ ↔ ∃𝑡 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)))
3	2	3com23 1144	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩ ↔ ∃𝑡 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)))
4	1, 3	anbi12d 644	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩) ↔ (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ ∃𝑡 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))))
5		reeanv 3238	. . 3 ⊢ (∃𝑦 ∈ (𝔼‘𝑁)∃𝑡 ∈ (𝔼‘𝑁)((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)) ↔ (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ ∃𝑡 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)))
6	4, 5	bitr4di 292	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑁)∃𝑡 ∈ (𝔼‘𝑁)((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))))
7		simpl1 1210	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
8		simpl3l 1247	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
9		simprr 785	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → 𝑡 ∈ (𝔼‘𝑁))
10		simprl 783	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → 𝑦 ∈ (𝔼‘𝑁))
11		simpl3r 1248	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → 𝐷 ∈ (𝔼‘𝑁))
12		simprll 791	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → 𝑦 Btwn ⟨𝐶, 𝐷⟩)
13		simprrl 793	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → 𝐷 Btwn ⟨𝐶, 𝑡⟩)
14	7, 8, 10, 11, 9, 12, 13	btwnexchand 36969	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → 𝑦 Btwn ⟨𝐶, 𝑡⟩)
15		simpl2l 1245	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
16		simpl2r 1246	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
17		simprrr 794	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)
18		simprlr 792	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)
19	7, 8, 9, 15, 16, 8, 10, 17, 18	cgrtrand 36936	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → ⟨𝐶, 𝑡⟩Cgr⟨𝐶, 𝑦⟩)
20	7, 8, 9, 10, 14, 19	endofsegidand 37029	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → 𝑡 = 𝑦)
21		opeq2 4844	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → ⟨𝐶, 𝑡⟩ = ⟨𝐶, 𝑦⟩)
22	21	breq2d 5118	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝐷 Btwn ⟨𝐶, 𝑡⟩ ↔ 𝐷 Btwn ⟨𝐶, 𝑦⟩))
23	21	breq1d 5116	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))
24	22, 23	anbi12d 644	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩) ↔ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩)))
25	24	anbi2d 642	. . . . . . 7 ⊢ (𝑡 = 𝑦 → (((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)) ↔ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))))
26	25	anbi2d 642	. . . . . 6 ⊢ (𝑡 = 𝑦 → ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) ↔ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩)))))
27		simprrl 793	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → 𝐷 Btwn ⟨𝐶, 𝑦⟩)
28	7, 11, 8, 10, 27	btwncomand 36958	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → 𝐷 Btwn ⟨𝑦, 𝐶⟩)
29		simprll 791	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → 𝑦 Btwn ⟨𝐶, 𝐷⟩)
30	7, 10, 8, 11, 29	btwncomand 36958	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → 𝑦 Btwn ⟨𝐷, 𝐶⟩)
31		btwnswapid 36960	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝐷 Btwn ⟨𝑦, 𝐶⟩ ∧ 𝑦 Btwn ⟨𝐷, 𝐶⟩) → 𝐷 = 𝑦))
32	7, 11, 10, 8, 31	syl13anc 1400	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) → ((𝐷 Btwn ⟨𝑦, 𝐶⟩ ∧ 𝑦 Btwn ⟨𝐷, 𝐶⟩) → 𝐷 = 𝑦))
33	32	adantr 486	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → ((𝐷 Btwn ⟨𝑦, 𝐶⟩ ∧ 𝑦 Btwn ⟨𝐷, 𝐶⟩) → 𝐷 = 𝑦))
34	28, 30, 33	mp2and 712	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → 𝐷 = 𝑦)
35		simprlr 792	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)
36		opeq2 4844	. . . . . . . . 9 ⊢ (𝐷 = 𝑦 → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑦⟩)
37	36	breq2d 5118	. . . . . . . 8 ⊢ (𝐷 = 𝑦 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
38	35, 37	syl5ibrcom 250	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → (𝐷 = 𝑦 → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩))
39	34, 38	mpd 16	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)
40	26, 39	biimtrdi 256	. . . . 5 ⊢ (𝑡 = 𝑦 → ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩))
41	20, 40	mpcom 39	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁))) ∧ ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩))) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)
42	41	exp31 425	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝑦 ∈ (𝔼‘𝑁) ∧ 𝑡 ∈ (𝔼‘𝑁)) → (((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)))
43	42	rexlimdvv 3222	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑦 ∈ (𝔼‘𝑁)∃𝑡 ∈ (𝔼‘𝑁)((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩))
44	6, 43	sylbid 243	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  ⟨cop 4838   class class class wbr 5106  ‘cfv 6558  ℕcn 12362  𝔼cee 29495   Btwn cbtwn 29496  Cgrccgr 29497   Seg_≤ csegle 37049

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

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-nel 3066  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-pss 4017  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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-n0 12634  df-z 12721  df-uz 12993  df-rp 13148  df-ico 13509  df-icc 13510  df-fz 13667  df-fzo 13816  df-seq 14172  df-exp 14232  df-hash 14502  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-clim 15684  df-sum 15883  df-ee 29498  df-btwn 29499  df-cgr 29500  df-ofs 36926  df-colinear 36982  df-ifs 36983  df-cgr3 36984  df-segle 37050

This theorem is used by:  colinbtwnle  37061