Theorem cgracgr 29140

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 2762	. . 3 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
3		iscgra.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		iscgra.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 742	. . 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 742	. . 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 742	. . 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 742	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑋 ∈ 𝑃)
12		eqid 2762	. . 3 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		simpllr 787	. . 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 742	. . 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 742	. . 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 742	. . 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 742	. . 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 28889	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
26	25	necomd 3012	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
27	1, 3, 23, 10, 6, 8, 4, 2, 24	hlln 28890	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴(LineG‘𝐺)𝐵))
28	1, 3, 2, 4, 8, 6, 10, 26, 27	lncom 28906	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐵(LineG‘𝐺)𝐴))
29	28	orcd 886	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐴) ∨ 𝐵 = 𝐴))
30	1, 2, 3, 4, 8, 6, 10, 29	colrot1 28839	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝐴(LineG‘𝐺)𝑋) ∨ 𝐴 = 𝑋))
31	30	ad3antrrr 742	. . 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 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
34		simplr 780	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
35		simpr1 1212	. . . . 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 28800	. . . 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 742	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐵 − 𝑋) = (𝐸 − 𝐷))
39		eqid 2762	. . . . . . 7 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
40		simpr2 1213	. . . . . . . . 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 28885	. . . . . . . . 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 235	. . . . . . . 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 1161	. . . . . . 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 28885	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(𝐾‘𝐵)𝐴 ↔ (𝑋 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))))
45	24, 44	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋))))
46	45	simp3d 1161	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))
47	46	orcomd 884	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝑋) ∨ 𝑋 ∈ (𝐵𝐼𝐴)))
48	47	ad3antrrr 742	. . . . . . 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 2768	. . . . . . . 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 28760	. . . . . . 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 2768	. . . . . . 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 28875	. . . . . 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 2768	. . . . 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 28760	. . . 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 28796	. . 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 28890	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐶(LineG‘𝐺)𝐵))
58	57	orcd 886	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ (𝐶(LineG‘𝐺)𝐵) ∨ 𝐶 = 𝐵))
59	1, 2, 3, 4, 32, 8, 17, 58	colrot1 28839	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑌) ∨ 𝐵 = 𝑌))
60	59	ad3antrrr 742	. . . . 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 28801	. . . . . 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 28885	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌(𝐾‘𝐵)𝐶 ↔ (𝑌 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))))
63	56, 62	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌))))
64	63	simp3d 1161	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))
65	64	orcomd 884	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
66	65	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
67		simpr3 1214	. . . . . . . . 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 28885	. . . . . . . . 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 235	. . . . . . . 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 1161	. . . . . . 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 742	. . . . . . 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 28875	. . . . . 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 28760	. . . . . 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 28796	. . . . 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 2768	. . . . 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 28802	. . . . 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 29135	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
80	79	ad3antrrr 742	. . . . 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 28848	. . . 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 28760	. . 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 742	. . 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 28848	. 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 29131	. . 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 235	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))
87	84, 86	r19.29vva 3224	1 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐷 − 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  ⟨“cs3 14886  Basecbs 17275  distcds 17325  TarskiGcstrkg 28707  Itvcitv 28713  LineGclng 28714  cgrGccgrg 28790  ≤Gcleg 28862  hlGchlg 28880  cgrAccgra 29129

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-oadd 8455  df-er 8692  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9894  df-card 9932  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-n0 12511  df-xnn0 12584  df-z 12598  df-uz 12869  df-fz 13542  df-fzo 13690  df-hash 14374  df-word 14558  df-concat 14615  df-s1 14641  df-s2 14892  df-s3 14893  df-trkgc 28728  df-trkgb 28729  df-trkgcb 28730  df-trkg 28733  df-cgrg 28791  df-leg 28863  df-hlg 28881  df-cgra 29130

This theorem is used by:  cgracom  29144  cgratr  29145  dfcgra2  29152  tgsas1  29182  tgasa1  29186