btwnconn1lem1

Description: Lemma for btwnconn1 . The next several lemmas introduce various properties of hypothetical points that end up eliminating alternatives to connectivity. We begin by showing a congruence property of those hypothetical points. (Contributed by Scott Fenton, 8-Oct-2013)

		Ref	Expression
	Assertion	btwnconn1lem1	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨B, c⟩ Cgr ⟨X, C⟩$

Proof

Step	Hyp	Ref	Expression
1		simp11	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to N \in ℕ$
2		simp13	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to B \in 𝔼 (N)$
3		simp22	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to D \in 𝔼 (N)$
4		simp23	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to c \in 𝔼 (N)$
5		simp33	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to X \in 𝔼 (N)$
6		simp31	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to d \in 𝔼 (N)$
7		simp21	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to C \in 𝔼 (N)$
8		simp12	$⊢ ((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \to A \in 𝔼 (N)$
9		simp1rr	$⊢ (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩))) \to B Btwn ⟨A, D⟩$
10	9	adantl	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to B Btwn ⟨A, D⟩$
11		simp2ll	$⊢ (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩))) \to D Btwn ⟨A, c⟩$
12	11	adantl	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to D Btwn ⟨A, c⟩$
13	1 8 2 3 4 10 12	btwnexch3and	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to D Btwn ⟨B, c⟩$
14		simp2rl	$⊢ (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩))) \to C Btwn ⟨A, d⟩$
15	14	adantl	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to C Btwn ⟨A, d⟩$
16		simp3rl	$⊢ (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩))) \to d Btwn ⟨A, X⟩$
17	16	adantl	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to d Btwn ⟨A, X⟩$
18	1 8 7 6 5 15 17	btwnexch3and	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to d Btwn ⟨C, X⟩$
19	1 6 7 5 18	btwncomand	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to d Btwn ⟨X, C⟩$
20		simp3rr	$⊢ (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩))) \to ⟨d, X⟩ Cgr ⟨D, B⟩$
21	20	adantl	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨d, X⟩ Cgr ⟨D, B⟩$
22	1 6 5 3 2 21	cgrcomand	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨D, B⟩ Cgr ⟨d, X⟩$
23	1 3 2 6 5 22	cgrcomlrand	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨B, D⟩ Cgr ⟨X, d⟩$
24		simp2lr	$⊢ (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩))) \to ⟨D, c⟩ Cgr ⟨C, D⟩$
25	24	adantl	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨D, c⟩ Cgr ⟨C, D⟩$
26		simp2rr	$⊢ (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩))) \to ⟨C, d⟩ Cgr ⟨C, D⟩$
27	26	adantl	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨C, d⟩ Cgr ⟨C, D⟩$
28	1 7 6 7 3 27	cgrcomland	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨d, C⟩ Cgr ⟨C, D⟩$
29	1 3 4 6 7 7 3 25 28	cgrtr3and	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨D, c⟩ Cgr ⟨d, C⟩$
30	1 2 3 4 5 6 7 13 19 23 29	cgrextendand	$⊢ (((N \in ℕ \land A \in 𝔼 (N) \land B \in 𝔼 (N)) \land (C \in 𝔼 (N) \land D \in 𝔼 (N) \land c \in 𝔼 (N)) \land (d \in 𝔼 (N) \land b \in 𝔼 (N) \land X \in 𝔼 (N))) \land (((A \neq B \land B \neq C) \land (B Btwn ⟨A, C⟩ \land B Btwn ⟨A, D⟩)) \land ((D Btwn ⟨A, c⟩ \land ⟨D, c⟩ Cgr ⟨C, D⟩) \land (C Btwn ⟨A, d⟩ \land ⟨C, d⟩ Cgr ⟨C, D⟩)) \land ((c Btwn ⟨A, b⟩ \land ⟨c, b⟩ Cgr ⟨C, B⟩) \land (d Btwn ⟨A, X⟩ \land ⟨d, X⟩ Cgr ⟨D, B⟩)))) \to ⟨B, c⟩ Cgr ⟨X, C⟩$

Metamath Proof Explorer

Theorem btwnconn1lem1

Proof