Theorem zerocgra 29242

Description: Zero angles are congruent. Zero angles, that is, angles of degree zero, can be expressed by stating that points 𝐴 and 𝐶 are on the same ray starting at 𝐵, that is, 𝐴(𝐾‘𝐵)𝐶. See also flatcgra 29243. (Contributed by Thierry Arnoux, 23-Aug-2026.)

Hypotheses

Ref	Expression
zerocgra.p	⊢ 𝑃 = (Base‘𝐺)
zerocgra.a	⊢ ∼ = (cgrA‘𝐺)
zerocgra.k	⊢ 𝐾 = (hlG‘𝐺)
zerocgra.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
zerocgra.1	⊢ (𝜑 → 𝐴(𝐾‘𝐵)𝐶)
zerocgra.2	⊢ (𝜑 → 𝐷(𝐾‘𝐸)𝐹)
zerocgra.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
zerocgra.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)

Assertion

Ref	Expression
zerocgra	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)

Proof of Theorem zerocgra

Dummy variables 𝑑 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zerocgra.a	. . . . . . . 8 ⊢ ∼ = (cgrA‘𝐺)
2	1	eqcomi 2769	. . . . . . 7 ⊢ (cgrA‘𝐺) = ∼
3	2	a1i 11	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → (cgrA‘𝐺) = ∼ )
4		zerocgra.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
5		eqid 2760	. . . . . . 7 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
6		zerocgra.k	. . . . . . 7 ⊢ 𝐾 = (hlG‘𝐺)
7		zerocgra.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8	7	ad6antr 749	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐺 ∈ TarskiG)
9		zerocgra.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10		zerocgra.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴(𝐾‘𝐵)𝐶)
11	4, 5, 6, 7, 9, 10	hlgrcl1 28977	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12	11	ad6antr 749	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐴 ∈ 𝑃)
13	9	ad6antr 749	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐵 ∈ 𝑃)
14	4, 5, 6, 7, 9, 10	hlgrcl2 28978	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	ad6antr 749	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐶 ∈ 𝑃)
16		zerocgra.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
17		zerocgra.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐷(𝐾‘𝐸)𝐹)
18	4, 5, 6, 7, 16, 17	hlgrcl1 28977	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
19	18	ad6antr 749	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐷 ∈ 𝑃)
20	16	ad6antr 749	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐸 ∈ 𝑃)
21	4, 5, 6, 7, 16, 17	hlgrcl2 28978	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
22	21	ad6antr 749	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐹 ∈ 𝑃)
23		simp-6r 800	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝑑 ∈ 𝑃)
24		simpllr 788	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝑓 ∈ 𝑃)
25		eqid 2760	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
26		eqid 2760	. . . . . . . 8 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
27		simp-4r 796	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴))
28	27	eqcomd 2766	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝑑))
29	4, 25, 5, 8, 13, 12, 20, 23, 28	tgcgrcomlr 28853	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → (𝐴(dist‘𝐺)𝐵) = (𝑑(dist‘𝐺)𝐸))
30		simpr 490	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶))
31	30	eqcomd 2766	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝑓))
32	10	ad6antr 749	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐴(𝐾‘𝐵)𝐶)
33		simp-5r 798	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝑑(𝐾‘𝐸)𝐷)
34		simplr 781	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝑓(𝐾‘𝐸)𝐹)
35	17	ad6antr 749	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐷(𝐾‘𝐸)𝐹)
36	4, 5, 6, 19, 22, 20, 8, 35	hlcomd 28981	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐹(𝐾‘𝐸)𝐷)
37	4, 5, 6, 24, 22, 19, 8, 20, 34, 36	hltr 28987	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝑓(𝐾‘𝐸)𝐷)
38	4, 5, 6, 24, 19, 20, 8, 37	hlcomd 28981	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝐷(𝐾‘𝐸)𝑓)
39	4, 5, 6, 23, 19, 24, 8, 20, 33, 38	hltr 28987	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → 𝑑(𝐾‘𝐸)𝑓)
40	4, 25, 6, 8, 13, 20, 32, 39, 31, 28	tghlsub 28997	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → (𝐶(dist‘𝐺)𝐴) = (𝑓(dist‘𝐺)𝑑))
41	4, 25, 26, 8, 12, 13, 15, 23, 20, 24, 29, 31, 40	trgcgr 28890	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑓”⟩)
42	4, 5, 6, 8, 12, 13, 15, 19, 20, 22, 23, 24, 41, 33, 34	iscgrad 29229	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
43	3, 42	breqdi 5118	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ 𝑓(𝐾‘𝐸)𝐹) ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
44	43	anasss 472	. . . 4 ⊢ ((((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) ∧ 𝑓 ∈ 𝑃) ∧ (𝑓(𝐾‘𝐸)𝐹 ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶))) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
45	4, 5, 6, 18, 21, 16, 7, 17	hlne2 28983	. . . . . 6 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
46	4, 5, 6, 11, 14, 9, 7, 10	hlne2 28983	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
47	46	necomd 3010	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
48	4, 5, 6, 16, 9, 14, 7, 21, 25, 45, 47	hlcgrex 28993	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝑓(𝐾‘𝐸)𝐹 ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)))
49	48	ad3antrrr 743	. . . 4 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) → ∃𝑓 ∈ 𝑃 (𝑓(𝐾‘𝐸)𝐹 ∧ (𝐸(dist‘𝐺)𝑓) = (𝐵(dist‘𝐺)𝐶)))
50	44, 49	r19.29a 3170	. . 3 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ 𝑑(𝐾‘𝐸)𝐷) ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
51	50	anasss 472	. 2 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝑃) ∧ (𝑑(𝐾‘𝐸)𝐷 ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴))) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
52	4, 5, 6, 18, 21, 16, 7, 17	hlne1 28982	. . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
53	4, 5, 6, 11, 14, 9, 7, 10	hlne1 28982	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
54	53	necomd 3010	. . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
55	4, 5, 6, 16, 9, 11, 7, 18, 25, 52, 54	hlcgrex 28993	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑃 (𝑑(𝐾‘𝐸)𝐷 ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)))
56	51, 55	r19.29a 3170	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   class class class wbr 5103  ‘cfv 6535  (class class class)co 7416  ⟨“cs3 14938  Basecbs 17326  distcds 17376  TarskiGcstrkg 28800  Itvcitv 28806  cgrGccgrg 28884  hlGchlg 28974  cgrAccgra 29225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-oadd 8466  df-er 8703  df-map 8835  df-pm 8836  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9931  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-2 12352  df-3 12353  df-n0 12554  df-xnn0 12627  df-z 12641  df-uz 12913  df-fz 13587  df-fzo 13735  df-hash 14420  df-word 14604  df-concat 14661  df-s1 14688  df-s2 14944  df-s3 14945  df-trkgc 28821  df-trkgb 28822  df-trkgcb 28823  df-trkg 28826  df-cgrg 28885  df-leg 28957  df-hlg 28975  df-cgra 29226

This theorem is used by:  angmgmaddeu2  29291  angmgmaddeu4  29293  angmgmaddeu6  29295  angmgmaddlid  29303  angmgmaddrid  29304