Theorem segleantisym 34344

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 34337	. . . 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 34338	. . . . 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 1124	. . . 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 630	. . 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 3292	. . 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 288	. 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 1189	. . . . . 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 1226	. . . . . 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 769	. . . . . 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 767	. . . . . 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 1227	. . . . . . 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 775	. . . . . . 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 777	. . . . . . 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 34255	. . . . . 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 1224	. . . . . . 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 1225	. . . . . . 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 778	. . . . . . 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 776	. . . . . . 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 34222	. . . . . 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 34315	. . . . 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 4802	. . . . . . . . . 10 ⊢ (𝑡 = 𝑦 → ⟨𝐶, 𝑡⟩ = ⟨𝐶, 𝑦⟩)
22	21	breq2d 5082	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝐷 Btwn ⟨𝐶, 𝑡⟩ ↔ 𝐷 Btwn ⟨𝐶, 𝑦⟩))
23	21	breq1d 5080	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩))
24	22, 23	anbi12d 630	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ((𝐷 Btwn ⟨𝐶, 𝑡⟩ ∧ ⟨𝐶, 𝑡⟩Cgr⟨𝐴, 𝐵⟩) ↔ (𝐷 Btwn ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩)))
25	24	anbi2d 628	. . . . . . 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 628	. . . . . 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 777	. . . . . . . . 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 34244	. . . . . . . 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 775	. . . . . . . . 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 34244	. . . . . . . 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 34246	. . . . . . . . . 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 1370	. . . . . . . . 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 695	. . . . . . 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 776	. . . . . . . 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 4802	. . . . . . . . 9 ⊢ (𝐷 = 𝑦 → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑦⟩)
37	36	breq2d 5082	. . . . . . . 8 ⊢ (𝐷 = 𝑦 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
38	35, 37	syl5ibrcom 246	. . . . . . 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	syl6bi 252	. . . . 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 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 239	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  ⟨cop 4564   class class class wbr 5070  ‘cfv 6418  ℕcn 11903  𝔼cee 27159   Btwn cbtwn 27160  Cgrccgr 27161   Seg_≤ csegle 34335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-ee 27162  df-btwn 27163  df-cgr 27164  df-ofs 34212  df-colinear 34268  df-ifs 34269  df-cgr3 34270  df-segle 34336

This theorem is referenced by:  colinbtwnle  34347