Theorem cgracgr 28874

Description: First direction of proposition 11.4 of [Schwabhauser] p. 95. Again, this is "half" of the proposition, i.e. only two additional points are used, while Schwabhauser has four. (Contributed by Thierry Arnoux, 31-Jul-2020.)

Hypotheses

Ref	Expression
iscgra.p	⊢ 𝑃 = (Base‘𝐺)
iscgra.i	⊢ 𝐼 = (Itv‘𝐺)
iscgra.k	⊢ 𝐾 = (hlG‘𝐺)
iscgra.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
iscgra.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
iscgra.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
iscgra.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
iscgra.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
iscgra.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
iscgra.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
cgrahl1.2	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
cgrahl1.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
cgracgr.m	⊢ − = (dist‘𝐺)
cgracgr.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
cgracgr.1	⊢ (𝜑 → 𝑋(𝐾‘𝐵)𝐴)
cgracgr.2	⊢ (𝜑 → 𝑌(𝐾‘𝐵)𝐶)
cgracgr.3	⊢ (𝜑 → (𝐵 − 𝑋) = (𝐸 − 𝐷))
cgracgr.4	⊢ (𝜑 → (𝐵 − 𝑌) = (𝐸 − 𝐹))

Assertion

Ref	Expression
cgracgr	⊢ (𝜑 → (𝑋 − 𝑌) = (𝐷 − 𝐹))

Proof of Theorem cgracgr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscgra.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2735	. . 3 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
3		iscgra.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		iscgra.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
6		iscgra.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐴 ∈ 𝑃)
8		iscgra.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐵 ∈ 𝑃)
10		cgrahl1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑋 ∈ 𝑃)
12		eqid 2735	. . 3 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		simpllr 776	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥 ∈ 𝑃)
14		iscgra.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
15	14	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃)
16		cgracgr.m	. . 3 ⊢ − = (dist‘𝐺)
17		cgracgr.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
18	17	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑌 ∈ 𝑃)
19		iscgra.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
20	19	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐷 ∈ 𝑃)
21		iscgra.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
22	21	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐹 ∈ 𝑃)
23		iscgra.k	. . . . . . . . 9 ⊢ 𝐾 = (hlG‘𝐺)
24		cgracgr.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑋(𝐾‘𝐵)𝐴)
25	1, 3, 23, 10, 6, 8, 4, 24	hlne2 28662	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
26	25	necomd 2985	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
27	1, 3, 23, 10, 6, 8, 4, 2, 24	hlln 28663	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴(LineG‘𝐺)𝐵))
28	1, 3, 2, 4, 8, 6, 10, 26, 27	lncom 28678	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐵(LineG‘𝐺)𝐴))
29	28	orcd 874	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐴) ∨ 𝐵 = 𝐴))
30	1, 2, 3, 4, 8, 6, 10, 29	colrot1 28615	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝐴(LineG‘𝐺)𝑋) ∨ 𝐴 = 𝑋))
31	30	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐵 ∈ (𝐴(LineG‘𝐺)𝑋) ∨ 𝐴 = 𝑋))
32		iscgra.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
33	32	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
34		simplr 769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
35		simpr1 1196	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
36	1, 16, 3, 12, 5, 7, 9, 33, 13, 15, 34, 35	cgr3simp1 28576	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐴 − 𝐵) = (𝑥 − 𝐸))
37		cgracgr.3	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝑋) = (𝐸 − 𝐷))
38	37	ad3antrrr 731	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐵 − 𝑋) = (𝐸 − 𝐷))
39		eqid 2735	. . . . . . 7 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
40		simpr2 1197	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥(𝐾‘𝐸)𝐷)
41	1, 3, 23, 13, 20, 15, 5	ishlg 28658	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥(𝐾‘𝐸)𝐷 ↔ (𝑥 ≠ 𝐸 ∧ 𝐷 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))))
42	40, 41	mpbid 232	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥 ≠ 𝐸 ∧ 𝐷 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥))))
43	42	simp3d 1145	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))
44	1, 3, 23, 10, 6, 8, 4	ishlg 28658	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(𝐾‘𝐵)𝐴 ↔ (𝑋 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))))
45	24, 44	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋))))
46	45	simp3d 1145	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))
47	46	orcomd 872	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝑋) ∨ 𝑋 ∈ (𝐵𝐼𝐴)))
48	47	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐴 ∈ (𝐵𝐼𝑋) ∨ 𝑋 ∈ (𝐵𝐼𝐴)))
49	36	eqcomd 2741	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥 − 𝐸) = (𝐴 − 𝐵))
50	1, 16, 3, 5, 13, 15, 7, 9, 49	tgcgrcomlr 28536	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐸 − 𝑥) = (𝐵 − 𝐴))
51	38	eqcomd 2741	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐸 − 𝐷) = (𝐵 − 𝑋))
52	1, 16, 3, 39, 5, 15, 13, 20, 9, 9, 7, 11, 43, 48, 50, 51	tgcgrsub2 28651	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥 − 𝐷) = (𝐴 − 𝑋))
53	52	eqcomd 2741	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐴 − 𝑋) = (𝑥 − 𝐷))
54	1, 16, 3, 5, 7, 11, 13, 20, 53	tgcgrcomlr 28536	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑋 − 𝐴) = (𝐷 − 𝑥))
55	1, 16, 12, 5, 7, 9, 11, 13, 15, 20, 36, 38, 54	trgcgr 28572	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝐷”⟩)
56		cgracgr.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑌(𝐾‘𝐵)𝐶)
57	1, 3, 23, 17, 32, 8, 4, 2, 56	hlln 28663	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐶(LineG‘𝐺)𝐵))
58	57	orcd 874	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ (𝐶(LineG‘𝐺)𝐵) ∨ 𝐶 = 𝐵))
59	1, 2, 3, 4, 32, 8, 17, 58	colrot1 28615	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑌) ∨ 𝐵 = 𝑌))
60	59	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑌) ∨ 𝐵 = 𝑌))
61	1, 16, 3, 12, 5, 7, 9, 33, 13, 15, 34, 35	cgr3simp2 28577	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐵 − 𝐶) = (𝐸 − 𝑦))
62	1, 3, 23, 17, 32, 8, 4	ishlg 28658	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌(𝐾‘𝐵)𝐶 ↔ (𝑌 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))))
63	56, 62	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌))))
64	63	simp3d 1145	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))
65	64	orcomd 872	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
66	65	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
67		simpr3 1198	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑦(𝐾‘𝐸)𝐹)
68	1, 3, 23, 34, 22, 15, 5	ishlg 28658	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑦(𝐾‘𝐸)𝐹 ↔ (𝑦 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))))
69	67, 68	mpbid 232	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑦 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦))))
70	69	simp3d 1145	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))
71		cgracgr.4	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑌) = (𝐸 − 𝐹))
72	71	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐵 − 𝑌) = (𝐸 − 𝐹))
73	1, 16, 3, 39, 5, 9, 33, 18, 15, 15, 34, 22, 66, 70, 61, 72	tgcgrsub2 28651	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐶 − 𝑌) = (𝑦 − 𝐹))
74	1, 16, 3, 5, 9, 18, 15, 22, 72	tgcgrcomlr 28536	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑌 − 𝐵) = (𝐹 − 𝐸))
75	1, 16, 12, 5, 9, 33, 18, 15, 34, 22, 61, 73, 74	trgcgr 28572	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → ⟨“𝐵𝐶𝑌”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝐹”⟩)
76	50	eqcomd 2741	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐵 − 𝐴) = (𝐸 − 𝑥))
77	1, 16, 3, 12, 5, 7, 9, 33, 13, 15, 34, 35	cgr3simp3 28578	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐶 − 𝐴) = (𝑦 − 𝑥))
78		cgrahl1.2	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
79	1, 3, 23, 4, 6, 8, 32, 19, 14, 21, 78	cgrane2 28869	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
80	79	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐵 ≠ 𝐶)
81	1, 2, 3, 5, 9, 33, 18, 12, 15, 34, 16, 7, 22, 13, 60, 75, 76, 77, 80	tgfscgr 28624	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑌 − 𝐴) = (𝐹 − 𝑥))
82	1, 16, 3, 5, 18, 7, 22, 13, 81	tgcgrcomlr 28536	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐴 − 𝑌) = (𝑥 − 𝐹))
83	25	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐴 ≠ 𝐵)
84	1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 16, 18, 20, 22, 31, 55, 82, 72, 83	tgfscgr 28624	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑋 − 𝑌) = (𝐷 − 𝐹))
85	1, 3, 23, 4, 6, 8, 32, 19, 14, 21	iscgra 28865	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)))
86	78, 85	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))
87	84, 86	r19.29vva 3195	1 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐷 − 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2930  ∃wrex 3059   class class class wbr 5074  ‘cfv 6487  (class class class)co 7356  ⟨“cs3 14793  Basecbs 17168  distcds 17218  TarskiGcstrkg 28483  Itvcitv 28489  LineGclng 28490  cgrGccgrg 28566  ≤Gcleg 28638  hlGchlg 28656  cgrAccgra 28863

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  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

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-oadd 8398  df-er 8632  df-map 8764  df-pm 8765  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-dju 9814  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-nn 12164  df-2 12233  df-3 12234  df-n0 12427  df-xnn0 12500  df-z 12514  df-uz 12778  df-fz 13451  df-fzo 13598  df-hash 14282  df-word 14465  df-concat 14522  df-s1 14548  df-s2 14799  df-s3 14800  df-trkgc 28504  df-trkgb 28505  df-trkgcb 28506  df-trkg 28509  df-cgrg 28567  df-leg 28639  df-hlg 28657  df-cgra 28864

This theorem is referenced by:  cgracom  28878  cgratr  28879  dfcgra2  28886  tgsas1  28910  tgasa1  28914