Theorem cgracom 26616

Description: Angle congruence commutes. Theorem 11.7 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)

Hypotheses

Ref	Expression
cgraid.p	⊢ 𝑃 = (Base‘𝐺)
cgraid.i	⊢ 𝐼 = (Itv‘𝐺)
cgraid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
cgraid.k	⊢ 𝐾 = (hlG‘𝐺)
cgraid.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
cgraid.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
cgraid.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
cgracom.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
cgracom.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
cgracom.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
cgracom.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Assertion

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

Proof of Theorem cgracom

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

Step	Hyp	Ref	Expression
1		cgraid.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2798	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2798	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
4		cgraid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐺 ∈ TarskiG)
6		cgracom.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
7	6	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐷 ∈ 𝑃)
8		cgracom.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
9	8	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐸 ∈ 𝑃)
10		cgracom.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
11	10	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐹 ∈ 𝑃)
12		simpllr 775	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑥 ∈ 𝑃)
13		cgraid.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
14	13	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐵 ∈ 𝑃)
15		simplr 768	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑦 ∈ 𝑃)
16		cgraid.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
17		simprlr 779	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷))
18	17	eqcomd 2804	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐸(dist‘𝐺)𝐷) = (𝐵(dist‘𝐺)𝑥))
19	1, 2, 16, 5, 9, 7, 14, 12, 18	tgcgrcomlr 26274	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐷(dist‘𝐺)𝐸) = (𝑥(dist‘𝐺)𝐵))
20		simprrr 781	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))
21	20	eqcomd 2804	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐸(dist‘𝐺)𝐹) = (𝐵(dist‘𝐺)𝑦))
22		cgraid.k	. . . . . . . 8 ⊢ 𝐾 = (hlG‘𝐺)
23		cgraid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
24	23	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐴 ∈ 𝑃)
25		cgraid.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
26	25	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐶 ∈ 𝑃)
27		cgracom.1	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
28	27	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
29		simprll 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑥(𝐾‘𝐵)𝐴)
30		simprrl 780	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑦(𝐾‘𝐵)𝐶)
31	1, 16, 22, 5, 24, 14, 26, 7, 9, 11, 28, 12, 2, 15, 29, 30, 17, 20	cgracgr 26612	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝑥(dist‘𝐺)𝑦) = (𝐷(dist‘𝐺)𝐹))
32	31	eqcomd 2804	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐷(dist‘𝐺)𝐹) = (𝑥(dist‘𝐺)𝑦))
33	1, 2, 16, 5, 7, 11, 12, 15, 32	tgcgrcomlr 26274	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐹(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝑥))
34	1, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 33	trgcgr 26310	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → ⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩)
35	34, 29, 30	3jca 1125	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾‘𝐵)𝐴 ∧ 𝑦(𝐾‘𝐵)𝐶))
36	1, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27	cgrane1 26606	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
37	1, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27	cgrane3 26608	. . . . 5 ⊢ (𝜑 → 𝐸 ≠ 𝐷)
38	1, 16, 22, 13, 8, 6, 4, 23, 2, 36, 37	hlcgrex 26410	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)))
39	1, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27	cgrane2 26607	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
40	39	necomd 3042	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
41	1, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27	cgrane4 26609	. . . . 5 ⊢ (𝜑 → 𝐸 ≠ 𝐹)
42	1, 16, 22, 13, 8, 10, 4, 25, 2, 40, 41	hlcgrex 26410	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))
43		reeanv 3320	. . . 4 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))) ↔ (∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))))
44	38, 42, 43	sylanbrc 586	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))))
45	35, 44	reximddv2 3237	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾‘𝐵)𝐴 ∧ 𝑦(𝐾‘𝐵)𝐶))
46	1, 16, 22, 4, 6, 8, 10, 23, 13, 25	iscgra 26603	. 2 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾‘𝐵)𝐴 ∧ 𝑦(𝐾‘𝐵)𝐶)))
47	45, 46	mpbird 260	1 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  ⟨“cs3 14195  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230  cgrGccgrg 26304  hlGchlg 26394  cgrAccgra 26601

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-s2 14201  df-s3 14202  df-trkgc 26242  df-trkgb 26243  df-trkgcb 26244  df-trkg 26247  df-cgrg 26305  df-leg 26377  df-hlg 26395  df-cgra 26602

This theorem is referenced by:  cgracol  26622  cgrancol  26623  dfcgra2  26624  tgasa1  26652  isoas  26658