Theorem cgracgr 27179

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 2738	. . 3 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
3		iscgra.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		iscgra.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 727	. . 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 727	. . 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 727	. . 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 727	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑋 ∈ 𝑃)
12		eqid 2738	. . 3 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		simpllr 773	. . 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 727	. . 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 727	. . 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 727	. . 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 727	. . 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 26967	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
26	25	necomd 2999	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
27	1, 3, 23, 10, 6, 8, 4, 2, 24	hlln 26968	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴(LineG‘𝐺)𝐵))
28	1, 3, 2, 4, 8, 6, 10, 26, 27	lncom 26983	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐵(LineG‘𝐺)𝐴))
29	28	orcd 870	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐴) ∨ 𝐵 = 𝐴))
30	1, 2, 3, 4, 8, 6, 10, 29	colrot1 26920	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝐴(LineG‘𝐺)𝑋) ∨ 𝐴 = 𝑋))
31	30	ad3antrrr 727	. . 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 727	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
34		simplr 766	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
35		simpr1 1193	. . . . 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 26881	. . . 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 727	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐵 − 𝑋) = (𝐸 − 𝐷))
39		eqid 2738	. . . . . . 7 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
40		simpr2 1194	. . . . . . . . 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 26963	. . . . . . . . 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 231	. . . . . . . 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 1143	. . . . . . 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 26963	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(𝐾‘𝐵)𝐴 ↔ (𝑋 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))))
45	24, 44	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋))))
46	45	simp3d 1143	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))
47	46	orcomd 868	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝑋) ∨ 𝑋 ∈ (𝐵𝐼𝐴)))
48	47	ad3antrrr 727	. . . . . . 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 2744	. . . . . . . 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 26841	. . . . . . 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 2744	. . . . . . 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 26956	. . . . . 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 2744	. . . . 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 26841	. . . 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 26877	. . 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 26968	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐶(LineG‘𝐺)𝐵))
58	57	orcd 870	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ (𝐶(LineG‘𝐺)𝐵) ∨ 𝐶 = 𝐵))
59	1, 2, 3, 4, 32, 8, 17, 58	colrot1 26920	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑌) ∨ 𝐵 = 𝑌))
60	59	ad3antrrr 727	. . . . 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 26882	. . . . . 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 26963	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌(𝐾‘𝐵)𝐶 ↔ (𝑌 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))))
63	56, 62	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌))))
64	63	simp3d 1143	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))
65	64	orcomd 868	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
66	65	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
67		simpr3 1195	. . . . . . . . 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 26963	. . . . . . . . 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 231	. . . . . . . 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 1143	. . . . . . 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 727	. . . . . . 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 26956	. . . . . 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 26841	. . . . . 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 26877	. . . . 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 2744	. . . . 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 26883	. . . . 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 27174	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
80	79	ad3antrrr 727	. . . . 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 26929	. . . 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 26841	. . 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 727	. . 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 26929	. 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 27170	. . 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 231	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))
87	84, 86	r19.29vva 3266	1 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐷 − 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  ⟨“cs3 14555  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794  LineGclng 26795  cgrGccgrg 26871  ≤Gcleg 26943  hlGchlg 26961  cgrAccgra 27168

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-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

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-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-tp 4566  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-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-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-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  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-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-s2 14561  df-s3 14562  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814  df-cgrg 26872  df-leg 26944  df-hlg 26962  df-cgra 27169

This theorem is referenced by:  cgracom  27183  cgratr  27184  dfcgra2  27191  tgsas1  27215  tgasa1  27219