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Theorem cgracgr 28572

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 2726	. . 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 2726	. . 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 28360	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
26	25	necomd 2990	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
27	1, 3, 23, 10, 6, 8, 4, 2, 24	hlln 28361	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴(LineG‘𝐺)𝐵))
28	1, 3, 2, 4, 8, 6, 10, 26, 27	lncom 28376	. . . . . 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 28313	. . . 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 1191	. . . . 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 28274	. . . 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 2726	. . . . . . 7 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
40		simpr2 1192	. . . . . . . . 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 28356	. . . . . . . . 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 1141	. . . . . . 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 28356	. . . . . . . . . . 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 1141	. . . . . . . . 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 2732	. . . . . . . 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 28234	. . . . . . 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 2732	. . . . . . 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 28349	. . . . . 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 2732	. . . . 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 28234	. . . 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 28270	. . 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 28361	. . . . . . . 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 28313	. . . . . 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 28275	. . . . . 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 28356	. . . . . . . . . . 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 1141	. . . . . . . . 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 1193	. . . . . . . . 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 28356	. . . . . . . . 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 1141	. . . . . . 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 28349	. . . . . 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 28234	. . . . . 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 28270	. . . . 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 2732	. . . . 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 28276	. . . . 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 28567	. . . . . 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 28322	. . . 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 28234	. . 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 28322	. 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 28563	. . 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 3207	1 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ⟨“cs3 14796 Basecbs 17150 distcds 17212 TarskiGcstrkg 28181 Itvcitv 28187 LineGclng 28188 cgrGccgrg 28264 ≤Gcleg 28336 hlGchlg 28354 cgrAccgra 28561

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14293 df-word 14468 df-concat 14524 df-s1 14549 df-s2 14802 df-s3 14803 df-trkgc 28202 df-trkgb 28203 df-trkgcb 28204 df-trkg 28207 df-cgrg 28265 df-leg 28337 df-hlg 28355 df-cgra 28562

This theorem is referenced by: cgracom 28576 cgratr 28577 dfcgra2 28584 tgsas1 28608 tgasa1 28612

W3C validator