Theorem btwnconn1lem8 34396

Description: Lemma for btwnconn1 34403. Now, we introduce the last three points used in the construction: 𝑃, 𝑄, and 𝑅 will turn out to be equal further down, and will provide us with the key to the final statement. We begin by establishing congruence of 𝑅𝑃 and 𝐸𝑑. (Contributed by Scott Fenton, 8-Oct-2013.)

Assertion

Ref

Expression

btwnconn1lem8

⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑅, 𝑃⟩Cgr⟨𝐸, 𝑑⟩)

Proof of Theorem btwnconn1lem8

Step	Hyp	Ref	Expression
1		simpr2l 1231	. . . 4 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → 𝐶 Btwn ⟨𝑑, 𝑅⟩)
2	1	ad2antll 726	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝑑, 𝑅⟩)
3		simpr1r 1230	. . . . . 6 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)
4	3	ad2antll 726	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)
5		simp11 1202	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
6		simp2l1 1271	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
7		simp31 1208	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
8		simp2r1 1274	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑑 ∈ (𝔼‘𝑁))
9		cgrcomlr 34300	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑑 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩ ↔ ⟨𝑃, 𝐶⟩Cgr⟨𝑑, 𝐶⟩))
10	5, 6, 7, 6, 8, 9	syl122anc 1378	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩ ↔ ⟨𝑃, 𝐶⟩Cgr⟨𝑑, 𝐶⟩))
11		cgrcom 34292	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (⟨𝑃, 𝐶⟩Cgr⟨𝑑, 𝐶⟩ ↔ ⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩))
12	5, 7, 6, 8, 6, 11	syl122anc 1378	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝑃, 𝐶⟩Cgr⟨𝑑, 𝐶⟩ ↔ ⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩))
13	10, 12	bitrd 278	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩ ↔ ⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩))
14	13	adantr 481	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩ ↔ ⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩))
15	4, 14	mpbid 231	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩)
16		simp33 1210	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑅 ∈ (𝔼‘𝑁))
17		simp2r3 1276	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝐸 ∈ (𝔼‘𝑁))
18		simp2l3 1273	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑐 ∈ (𝔼‘𝑁))
19		simpr1l 1229	. . . . . . . 8 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → 𝐶 Btwn ⟨𝑐, 𝑃⟩)
20	19	ad2antll 726	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝑐, 𝑃⟩)
21	5, 6, 18, 7, 20	btwncomand 34317	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝑃, 𝑐⟩)
22		simprll 776	. . . . . . 7 ⊢ (((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))) → 𝐸 Btwn ⟨𝐶, 𝑐⟩)
23	22	adantl 482	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐸 Btwn ⟨𝐶, 𝑐⟩)
24		btwnintr 34321	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐸 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁))) → ((𝐶 Btwn ⟨𝑃, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐶, 𝑐⟩) → 𝐶 Btwn ⟨𝑃, 𝐸⟩))
25	5, 7, 6, 17, 18, 24	syl122anc 1378	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((𝐶 Btwn ⟨𝑃, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐶, 𝑐⟩) → 𝐶 Btwn ⟨𝑃, 𝐸⟩))
26	25	adantr 481	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ((𝐶 Btwn ⟨𝑃, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐶, 𝑐⟩) → 𝐶 Btwn ⟨𝑃, 𝐸⟩))
27	21, 23, 26	mp2and 696	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝑃, 𝐸⟩)
28		simpr2r 1232	. . . . . 6 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩)
29	28	ad2antll 726	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩)
30	5, 8, 6, 16, 7, 6, 17, 2, 27, 15, 29	cgrextendand 34311	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑑, 𝑅⟩Cgr⟨𝑃, 𝐸⟩)
31		brcgr3 34348	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ↔ (⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩ ∧ ⟨𝑑, 𝑅⟩Cgr⟨𝑃, 𝐸⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩)))
32	5, 8, 6, 16, 7, 6, 17, 31	syl133anc 1392	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ↔ (⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩ ∧ ⟨𝑑, 𝑅⟩Cgr⟨𝑃, 𝐸⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩)))
33	32	adantr 481	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ↔ (⟨𝑑, 𝐶⟩Cgr⟨𝑃, 𝐶⟩ ∧ ⟨𝑑, 𝑅⟩Cgr⟨𝑃, 𝐸⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩)))
34	15, 30, 29, 33	mpbir3and 1341	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩)
35	5, 8, 7	cgrrflx2d 34286	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩)
36	35	adantr 481	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩)
37	36, 4	jca 512	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩))
38	2, 34, 37	3jca 1127	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ∧ (⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)))
39		simp1 1135	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
40		simp2l 1198	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)))
41		simp2r 1199	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)))
42	39, 40, 41	3jca 1127	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))))
43		simpl 483	. . . . 5 ⊢ (((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))) → (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))))
44		simprl 768	. . . . 5 ⊢ (((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))) → (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))
45	43, 44	jca 512	. . . 4 ⊢ (((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))) → ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩)))
46		btwnconn1lem7 34395	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))) → 𝐶 ≠ 𝑑)
47	42, 45, 46	syl2an 596	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 ≠ 𝑑)
48	47	necomd 2999	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑑 ≠ 𝐶)
49		brofs2 34379	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑑 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑅 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝑑 ∈ (𝔼‘𝑁))) → (⟨⟨𝑑, 𝐶⟩, ⟨𝑅, 𝑃⟩⟩ OuterFiveSeg ⟨⟨𝑃, 𝐶⟩, ⟨𝐸, 𝑑⟩⟩ ↔ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ∧ (⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩))))
50	49	anbi1d 630	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑑 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑅 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝑑 ∈ (𝔼‘𝑁))) → ((⟨⟨𝑑, 𝐶⟩, ⟨𝑅, 𝑃⟩⟩ OuterFiveSeg ⟨⟨𝑃, 𝐶⟩, ⟨𝐸, 𝑑⟩⟩ ∧ 𝑑 ≠ 𝐶) ↔ ((𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ∧ (⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)) ∧ 𝑑 ≠ 𝐶)))
51		5segofs 34308	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑑 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑅 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝑑 ∈ (𝔼‘𝑁))) → ((⟨⟨𝑑, 𝐶⟩, ⟨𝑅, 𝑃⟩⟩ OuterFiveSeg ⟨⟨𝑃, 𝐶⟩, ⟨𝐸, 𝑑⟩⟩ ∧ 𝑑 ≠ 𝐶) → ⟨𝑅, 𝑃⟩Cgr⟨𝐸, 𝑑⟩))
52	50, 51	sylbird 259	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑑 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑅 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝑑 ∈ (𝔼‘𝑁))) → (((𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ∧ (⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)) ∧ 𝑑 ≠ 𝐶) → ⟨𝑅, 𝑃⟩Cgr⟨𝐸, 𝑑⟩))
53	5, 8, 6, 16, 7, 7, 6, 17, 8, 52	syl333anc 1401	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (((𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ∧ (⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)) ∧ 𝑑 ≠ 𝐶) → ⟨𝑅, 𝑃⟩Cgr⟨𝐸, 𝑑⟩))
54	53	adantr 481	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (((𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝑑, ⟨𝐶, 𝑅⟩⟩Cgr3⟨𝑃, ⟨𝐶, 𝐸⟩⟩ ∧ (⟨𝑑, 𝑃⟩Cgr⟨𝑃, 𝑑⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)) ∧ 𝑑 ≠ 𝐶) → ⟨𝑅, 𝑃⟩Cgr⟨𝐸, 𝑑⟩))
55	38, 48, 54	mp2and 696	1 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑅, 𝑃⟩Cgr⟨𝐸, 𝑑⟩)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   ∈ wcel 2106   ≠ wne 2943  ⟨cop 4567   class class class wbr 5074  ‘cfv 6433  ℕcn 11973  𝔼cee 27256   Btwn cbtwn 27257  Cgrccgr 27258   OuterFiveSeg cofs 34284  Cgr3ccgr3 34338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-ee 27259  df-btwn 27260  df-cgr 27261  df-ofs 34285  df-ifs 34342  df-cgr3 34343

This theorem is referenced by:  btwnconn1lem9  34397  btwnconn1lem10  34398  btwnconn1lem11  34399