Theorem segleantisym 36478

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 36471	. . . 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 36472	. . . . 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 1142	. . . 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 643	. . 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 3237	. . 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 1208	. . . . . 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 1245	. . . . . 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 784	. . . . . 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 782	. . . . . 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 1246	. . . . . . 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 790	. . . . . . 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 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⟩))) → 𝐷 Btwn ⟨𝐶, 𝑡⟩)
14	7, 8, 10, 11, 9, 12, 13	btwnexchand 36389	. . . . . 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 1243	. . . . . . 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 1244	. . . . . . 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 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⟩))) → ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩)
18		simprlr 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⟩))) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)
19	7, 8, 9, 15, 16, 8, 10, 17, 18	cgrtrand 36356	. . . . . 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 36449	. . . . 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 4835	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → ⟨𝐶, 𝑡⟩ = ⟨𝐶, 𝑦⟩)
22	21	breq2d 5117	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝐷 Btwn ⟨𝐶, 𝑡⟩ ↔ 𝐷 Btwn ⟨𝐶, 𝑦⟩))
23	21	breq1d 5115	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))
24	22, 23	anbi12d 643	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩) ↔ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩)))
25	24	anbi2d 641	. . . . . . 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 641	. . . . . 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 792	. . . . . . . . 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 36378	. . . . . . . 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 790	. . . . . . . . 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 36378	. . . . . . . 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 36380	. . . . . . . . . 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 1395	. . . . . . . . 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 485	. . . . . . . 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 711	. . . . . . 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 791	. . . . . . . 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 4835	. . . . . . . . 9 ⊢ (𝐷 = 𝑦 → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑦⟩)
37	36	breq2d 5117	. . . . . . . 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 424	. . 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 3221	. 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⟩))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  ⟨cop 4591   class class class wbr 5105  ‘cfv 6525  ℕcn 12224  𝔼cee 29146   Btwn cbtwn 29147  Cgrccgr 29148   Seg_≤ csegle 36469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-rp 13008  df-ico 13369  df-icc 13370  df-fz 13527  df-fzo 13674  df-seq 14029  df-exp 14089  df-hash 14358  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-clim 15529  df-sum 15728  df-ee 29149  df-btwn 29150  df-cgr 29151  df-ofs 36346  df-colinear 36402  df-ifs 36403  df-cgr3 36404  df-segle 36470

This theorem is referenced by:  colinbtwnle  36481