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Theorem flatcgra 28644

Description: Flat angles are congruent. (Contributed by Thierry Arnoux, 13-Feb-2023.)

**Hypotheses**
Ref	Expression
cgracol.p	⊢ 𝑃 = (Base‘𝐺)
cgracol.i	⊢ 𝐼 = (Itv‘𝐺)
cgracol.m	⊢ − = (dist‘𝐺)
cgracol.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
cgracol.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
cgracol.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
cgracol.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
cgracol.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
cgracol.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
cgracol.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
flatcgra.1	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
flatcgra.2	⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))
flatcgra.3	⊢ (𝜑 → 𝐴 ≠ 𝐵)
flatcgra.4	⊢ (𝜑 → 𝐶 ≠ 𝐵)
flatcgra.5	⊢ (𝜑 → 𝐷 ≠ 𝐸)
flatcgra.6	⊢ (𝜑 → 𝐹 ≠ 𝐸)

**Assertion**
Ref	Expression
flatcgra	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Proof of Theorem flatcgra Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cgracol.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		cgracol.m	. . . . 5 ⊢ − = (dist‘𝐺)
3		eqid 2725	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
4		cgracol.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐺 ∈ TarskiG)
6		cgracol.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐴 ∈ 𝑃)
8		cgracol.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐵 ∈ 𝑃)
10		cgracol.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐶 ∈ 𝑃)
12		simpllr 774	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑥 ∈ 𝑃)
13		cgracol.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
14	13	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ∈ 𝑃)
15		simplr 767	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑦 ∈ 𝑃)
16		cgracol.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
17		simprlr 778	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐸 − 𝑥) = (𝐵 − 𝐴))
18	1, 2, 16, 5, 14, 12, 9, 7, 17	tgcgrcomlr 28300	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥 − 𝐸) = (𝐴 − 𝐵))
19	18	eqcomd 2731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐴 − 𝐵) = (𝑥 − 𝐸))
20		simprrr 780	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐸 − 𝑦) = (𝐵 − 𝐶))
21	20	eqcomd 2731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐵 − 𝐶) = (𝐸 − 𝑦))
22		cgracol.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
23	22	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐹 ∈ 𝑃)
24		cgracol.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
25	24	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐷 ∈ 𝑃)
26		flatcgra.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
27	26	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐹 ≠ 𝐸)
28		flatcgra.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
29	28	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐷 ≠ 𝐸)
30		flatcgra.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))
31	1, 2, 16, 4, 24, 13, 22, 30	tgbtwncom 28308	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (𝐹𝐼𝐷))
32	31	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ∈ (𝐹𝐼𝐷))
33		simprll 777	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ∈ (𝐹𝐼𝑥))
34		simprrl 779	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ∈ (𝐷𝐼𝑦))
35	1, 16, 5, 23, 14, 25, 12, 15, 27, 29, 32, 33, 34	tgbtwnconn22 28399	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ∈ (𝑥𝐼𝑦))
36		flatcgra.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
37	36	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐵 ∈ (𝐴𝐼𝐶))
38	1, 2, 16, 5, 12, 14, 15, 7, 9, 11, 35, 37, 18, 20	tgcgrextend 28305	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥 − 𝑦) = (𝐴 − 𝐶))
39	38	eqcomd 2731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐴 − 𝐶) = (𝑥 − 𝑦))
40	1, 2, 16, 5, 7, 11, 12, 15, 39	tgcgrcomlr 28300	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐶 − 𝐴) = (𝑦 − 𝑥))
41	1, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 40	trgcgr 28336	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
42	17	eqcomd 2731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐵 − 𝐴) = (𝐸 − 𝑥))
43		flatcgra.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
44	43	necomd 2986	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
45	44	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐵 ≠ 𝐴)
46	1, 2, 16, 5, 9, 7, 14, 12, 42, 45	tgcgrneq 28303	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ≠ 𝑥)
47	46	necomd 2986	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑥 ≠ 𝐸)
48	1, 16, 5, 23, 14, 12, 25, 27, 33, 32	tgbtwnconn2 28396	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))
49	47, 29, 48	3jca 1125	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥 ≠ 𝐸 ∧ 𝐷 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥))))
50		eqid 2725	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
51	1, 16, 50, 12, 25, 14, 5	ishlg 28422	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥((hlG‘𝐺)‘𝐸)𝐷 ↔ (𝑥 ≠ 𝐸 ∧ 𝐷 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))))
52	49, 51	mpbird 256	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑥((hlG‘𝐺)‘𝐸)𝐷)
53		flatcgra.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
54	53	necomd 2986	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
55	54	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐵 ≠ 𝐶)
56	1, 2, 16, 5, 9, 11, 14, 15, 21, 55	tgcgrneq 28303	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ≠ 𝑦)
57	56	necomd 2986	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑦 ≠ 𝐸)
58	30	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ∈ (𝐷𝐼𝐹))
59	1, 16, 5, 25, 14, 15, 23, 29, 34, 58	tgbtwnconn2 28396	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))
60	57, 27, 59	3jca 1125	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑦 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦))))
61	1, 16, 50, 15, 23, 14, 5	ishlg 28422	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑦((hlG‘𝐺)‘𝐸)𝐹 ↔ (𝑦 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))))
62	60, 61	mpbird 256	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑦((hlG‘𝐺)‘𝐸)𝐹)
63	41, 52, 62	3jca 1125	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹))
64	1, 2, 16, 4, 22, 13, 8, 6	axtgsegcon 28284	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)))
65	1, 2, 16, 4, 24, 13, 8, 10	axtgsegcon 28284	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))
66		reeanv 3217	. . . 4 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))) ↔ (∃𝑥 ∈ 𝑃 (𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ ∃𝑦 ∈ 𝑃 (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))))
67	64, 65, 66	sylanbrc 581	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))))
68	63, 67	reximddv2 3203	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹))
69	1, 16, 50, 4, 6, 8, 10, 24, 13, 22	iscgra 28629	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)))
70	68, 69	mpbird 256	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 class class class wbr 5141 ‘cfv 6541 (class class class)co 7414 ⟨“cs3 14823 Basecbs 17177 distcds 17239 TarskiGcstrkg 28247 Itvcitv 28253 cgrGccgrg 28330 hlGchlg 28420 cgrAccgra 28627

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 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-oadd 8487 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-concat 14551 df-s1 14576 df-s2 14829 df-s3 14830 df-trkgc 28268 df-trkgb 28269 df-trkgcb 28270 df-trkg 28273 df-cgrg 28331 df-hlg 28421 df-cgra 28628

This theorem is referenced by: (None)

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