Theorem segleantisym 36318

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 36311	. . . 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 36312	. . . . 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 1127	. . . 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 633	. . 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 3210	. . 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 289	. 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 1193	. . . . . 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 1230	. . . . . 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 773	. . . . . 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 771	. . . . . 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 1231	. . . . . . 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 779	. . . . . . 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 781	. . . . . . 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 36229	. . . . . 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 1228	. . . . . . 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 1229	. . . . . . 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 782	. . . . . . 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 780	. . . . . . 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 36196	. . . . . 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 36289	. . . . 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 4818	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → ⟨𝐶, 𝑡⟩ = ⟨𝐶, 𝑦⟩)
22	21	breq2d 5098	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝐷 Btwn ⟨𝐶, 𝑡⟩ ↔ 𝐷 Btwn ⟨𝐶, 𝑦⟩))
23	21	breq1d 5096	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))
24	22, 23	anbi12d 633	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩) ↔ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩)))
25	24	anbi2d 631	. . . . . . 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 631	. . . . . 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 781	. . . . . . . . 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 36218	. . . . . . . 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 779	. . . . . . . . 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 36218	. . . . . . . 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 36220	. . . . . . . . . 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 1375	. . . . . . . . 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 480	. . . . . . . 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 700	. . . . . . 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 780	. . . . . . . 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 4818	. . . . . . . . 9 ⊢ (𝐷 = 𝑦 → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑦⟩)
37	36	breq2d 5098	. . . . . . . 8 ⊢ (𝐷 = 𝑦 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
38	35, 37	syl5ibrcom 247	. . . . . . 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 15	. . . . . 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 253	. . . . 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 38	. . . 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 419	. . 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 3194	. 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 240	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ⟨cop 4574   class class class wbr 5086  ‘cfv 6490  ℕcn 12163  𝔼cee 28975   Btwn cbtwn 28976  Cgrccgr 28977   Seg_≤ csegle 36309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-inf2 9551  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-oi 9416  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12164  df-2 12233  df-3 12234  df-n0 12427  df-z 12514  df-uz 12778  df-rp 12932  df-ico 13293  df-icc 13294  df-fz 13451  df-fzo 13598  df-seq 13953  df-exp 14013  df-hash 14282  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15439  df-sum 15638  df-ee 28978  df-btwn 28979  df-cgr 28980  df-ofs 36186  df-colinear 36242  df-ifs 36243  df-cgr3 36244  df-segle 36310

This theorem is referenced by:  colinbtwnle  36321