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Theorem cgracom 28646

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 2728	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2728	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
4		cgraid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 728	. . . . 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 728	. . . . 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 728	. . . . 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 728	. . . . 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 774	. . . . 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 728	. . . . 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 767	. . . . 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 778	. . . . . . 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 2734	. . . . . 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 28304	. . . . 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 780	. . . . . 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 2734	. . . . 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 728	. . . . . . . 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 728	. . . . . . . 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 728	. . . . . . . 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 777	. . . . . . . 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 779	. . . . . . . 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 28642	. . . . . . 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 2734	. . . . . 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 28304	. . . . 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 28340	. . . 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 28636	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
37	1, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27	cgrane3 28638	. . . . 5 ⊢ (𝜑 → 𝐸 ≠ 𝐷)
38	1, 16, 22, 13, 8, 6, 4, 23, 2, 36, 37	hlcgrex 28440	. . . 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 28637	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
40	39	necomd 2993	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
41	1, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27	cgrane4 28639	. . . . 5 ⊢ (𝜑 → 𝐸 ≠ 𝐹)
42	1, 16, 22, 13, 8, 10, 4, 25, 2, 40, 41	hlcgrex 28440	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))
43		reeanv 3224	. . . 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 581	. . 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 3210	. 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 28633	. 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 256	1 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ⟨“cs3 14833 Basecbs 17187 distcds 17249 TarskiGcstrkg 28251 Itvcitv 28257 cgrGccgrg 28334 hlGchlg 28424 cgrAccgra 28631

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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 df-s2 14839 df-s3 14840 df-trkgc 28272 df-trkgb 28273 df-trkgcb 28274 df-trkg 28277 df-cgrg 28335 df-leg 28407 df-hlg 28425 df-cgra 28632

This theorem is referenced by: cgracol 28652 cgrancol 28653 dfcgra2 28654 tgasa1 28682 isoas 28688

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