Theorem segleantisym 36344

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 36337	. . . 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 36338	. . . . 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 1132	. . . 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 638	. . 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 3212	. . 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 290	. 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 1198	. . . . . 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 1235	. . . . . 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 778	. . . . . 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 776	. . . . . 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 1236	. . . . . . 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 784	. . . . . . 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 786	. . . . . . 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 36255	. . . . . 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 1233	. . . . . . 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 1234	. . . . . . 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 787	. . . . . . 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 785	. . . . . . 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 36222	. . . . . 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 36315	. . . . 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 4812	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → ⟨𝐶, 𝑡⟩ = ⟨𝐶, 𝑦⟩)
22	21	breq2d 5091	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝐷 Btwn ⟨𝐶, 𝑡⟩ ↔ 𝐷 Btwn ⟨𝐶, 𝑦⟩))
23	21	breq1d 5089	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))
24	22, 23	anbi12d 638	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩) ↔ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩)))
25	24	anbi2d 636	. . . . . . 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 636	. . . . . 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 786	. . . . . . . . 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 36244	. . . . . . . 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 784	. . . . . . . . 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 36244	. . . . . . . 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 36246	. . . . . . . . . 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 1380	. . . . . . . . 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 481	. . . . . . . 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 705	. . . . . . 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 785	. . . . . . . 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 4812	. . . . . . . . 9 ⊢ (𝐷 = 𝑦 → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑦⟩)
37	36	breq2d 5091	. . . . . . . 8 ⊢ (𝐷 = 𝑦 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
38	35, 37	syl5ibrcom 248	. . . . . . 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 254	. . . . 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 420	. . 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 3196	. 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 241	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  ⟨cop 4568   class class class wbr 5079  ‘cfv 6492  ℕcn 12172  𝔼cee 28981   Btwn cbtwn 28982  Cgrccgr 28983   Seg_≤ csegle 36335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-oi 9422  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-rp 12941  df-ico 13302  df-icc 13303  df-fz 13460  df-fzo 13607  df-seq 13962  df-exp 14022  df-hash 14291  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-clim 15448  df-sum 15647  df-ee 28984  df-btwn 28985  df-cgr 28986  df-ofs 36212  df-colinear 36268  df-ifs 36269  df-cgr3 36270  df-segle 36336

This theorem is referenced by:  colinbtwnle  36347