Theorem zerocgra 29206

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 29207. (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 2771	. . . . . . 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 2762	. . . . . . 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 28941	. . . . . . . 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 28942	. . . . . . . 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 28941	. . . . . . . 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 28942	. . . . . . . 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 2762	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
26		eqid 2762	. . . . . . . 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 2768	. . . . . . . . 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 28817	. . . . . . . 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 2768	. . . . . . . 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 28945	. . . . . . . . . . . 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 28951	. . . . . . . . . . 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 28945	. . . . . . . . . 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 28951	. . . . . . . . 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 28961	. . . . . . . 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 28854	. . . . . . 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 29193	. . . . . 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 5122	. . . . 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 28947	. . . . . 6 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
46	4, 5, 6, 11, 14, 9, 7, 10	hlne2 28947	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
47	46	necomd 3012	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
48	4, 5, 6, 16, 9, 14, 7, 21, 25, 45, 47	hlcgrex 28957	. . . . 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 3172	. . 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 28946	. . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
53	4, 5, 6, 11, 14, 9, 7, 10	hlne1 28946	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
54	53	necomd 3012	. . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
55	4, 5, 6, 16, 9, 11, 7, 18, 25, 52, 54	hlcgrex 28957	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑃 (𝑑(𝐾‘𝐸)𝐷 ∧ (𝐸(dist‘𝐺)𝑑) = (𝐵(dist‘𝐺)𝐴)))
56	51, 55	r19.29a 3172	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  ⟨“cs3 14913  Basecbs 17303  distcds 17353  TarskiGcstrkg 28764  Itvcitv 28770  cgrGccgrg 28848  hlGchlg 28938  cgrAccgra 29189

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-oadd 8462  df-er 8699  df-map 8831  df-pm 8832  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9909  df-card 9947  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-n0 12530  df-xnn0 12603  df-z 12617  df-uz 12889  df-fz 13562  df-fzo 13710  df-hash 14395  df-word 14579  df-concat 14636  df-s1 14663  df-s2 14919  df-s3 14920  df-trkgc 28785  df-trkgb 28786  df-trkgcb 28787  df-trkg 28790  df-cgrg 28849  df-leg 28921  df-hlg 28939  df-cgra 29190

This theorem is used by:  angmndaddeu2  29251  angmndaddeu4  29253  angmndaddeu6  29255