Theorem zerocgra 29358

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 29359. (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 2772	. . . . . . 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 2763	. . . . . . 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 29093	. . . . . . . 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 29094	. . . . . . . 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 29093	. . . . . . . 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 29094	. . . . . . . 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 2763	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
26		eqid 2763	. . . . . . . 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 2769	. . . . . . . . 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 28969	. . . . . . . 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 2769	. . . . . . . 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 29097	. . . . . . . . . . . 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 29103	. . . . . . . . . . 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 29097	. . . . . . . . . 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 29103	. . . . . . . . 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 29113	. . . . . . . 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 29006	. . . . . . 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 29345	. . . . . 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 5121	. . . . 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 29099	. . . . . 6 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
46	4, 5, 6, 11, 14, 9, 7, 10	hlne2 29099	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
47	46	necomd 3014	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
48	4, 5, 6, 16, 9, 14, 7, 21, 25, 45, 47	hlcgrex 29109	. . . . 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 3174	. . 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 29098	. . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
53	4, 5, 6, 11, 14, 9, 7, 10	hlne1 29098	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
54	53	necomd 3014	. . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
55	4, 5, 6, 16, 9, 11, 7, 18, 25, 52, 54	hlcgrex 29109	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑃 (𝑑(𝐾‘𝐸)𝐷 ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)))
56	51, 55	r19.29a 3174	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440  ⟨“cs3 15018  Basecbs 17413  distcds 17463  TarskiGcstrkg 28916  Itvcitv 28922  cgrGccgrg 29000  hlGchlg 29090  cgrAccgra 29341

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-oadd 8505  df-er 8742  df-map 8874  df-pm 8875  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813  df-hash 14499  df-word 14683  df-concat 14740  df-s1 14767  df-s2 15024  df-s3 15025  df-trkgc 28937  df-trkgb 28938  df-trkgcb 28939  df-trkg 28942  df-cgrg 29001  df-leg 29073  df-hlg 29091  df-cgra 29342

This theorem is used by:  angmgmaddeu2  29407  angmgmaddeu4  29409  angmgmaddeu6  29411  angmgmaddlid  29419  angmgmaddrid  29420