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Theorem symquadmid 29309

Description: In a symmetrical quadrilateral, the midpoints of the diagonals coincide. Corollary of Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 20-Jul-2026.)

**Hypotheses**
Ref	Expression
symquadmid.p	⊢ 𝑃 = (Base‘𝐺)
symquadmid.d	⊢ − = (dist‘𝐺)
symquadmid.i	⊢ 𝐼 = (Itv‘𝐺)
symquadmid.l	⊢ 𝐿 = (LineG‘𝐺)
symquadmid.m	⊢ 𝑀 = (midG‘𝐺)
symquadmid.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}
symquadmid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
symquadmid.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
symquadmid.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
symquadmid.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
symquadmid.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
symquadmid.2	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
symquadmid.3	⊢ (𝜑 → 𝑌 ≠ 𝑊)
symquadmid.4	⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
symquadmid.5	⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
symquadmid.6	⊢ (𝜑 → 𝑌𝑂𝑊)

**Assertion**
Ref	Expression
symquadmid	⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))

Distinct variable groups: 𝐼,𝑎,𝑏 𝐿,𝑎,𝑏 𝑡,𝑀 𝑃,𝑎,𝑏 𝑡,𝑊 𝑋,𝑎,𝑏,𝑡 𝑡,𝑌 𝑍,𝑎,𝑏,𝑡 𝜑,𝑡 Allowed substitution hints: 𝜑(𝑎, 𝑏) 𝑃(𝑡) 𝐺(𝑡, 𝑎, 𝑏) 𝐼(𝑡) 𝐿(𝑡) 𝑀(𝑎, 𝑏) − (𝑡, 𝑎, 𝑏) 𝑂(𝑡, 𝑎, 𝑏) 𝑊(𝑎, 𝑏) 𝑌(𝑎, 𝑏)

**Proof of Theorem symquadmid**
Step	Hyp	Ref	Expression
1		symquadmid.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
2		symquadmid.d	. . . . . 6 ⊢ − = (dist‘𝐺)
3		symquadmid.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
4		symquadmid.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2760	. . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		symquadmid.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝐺 ∈ TarskiG)
8		eqid 2760	. . . . . 6 ⊢ ((pInvG‘𝐺)‘𝑡) = ((pInvG‘𝐺)‘𝑡)
9		symquadmid.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 ∈ 𝑃)
11		symquadmid.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
12	11	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 ∈ 𝑃)
13		symquadmid.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
14	13	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑍 ∈ 𝑃)
15		symquadmid.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
16	15	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑊 ∈ 𝑃)
17		symquadmid.2	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
18	1, 3, 4, 6, 9, 11, 13, 17	ncolne2 29099	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
19	1, 3, 4, 6, 9, 13, 18	tgelrnln 29103	. . . . . . . 8 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
20	19	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋𝐿𝑍) ∈ ran 𝐿)
21		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑋𝐿𝑍))
22	1, 4, 3, 7, 20, 21	tglnpt 29017	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ 𝑃)
23	17	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
24		symquadmid.3	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
25	24	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 ≠ 𝑊)
26		symquadmid.4	. . . . . . 7 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
27	26	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋 − 𝑌) = (𝑍 − 𝑊))
28		symquadmid.5	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
29	28	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌 − 𝑍) = (𝑊 − 𝑋))
30	21	orcd 887	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑡 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
31		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑌𝐼𝑊))
32	1, 3, 4, 7, 12, 16, 22, 25, 31	btwnlng1 29092	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑌𝐿𝑊))
33	32	orcd 887	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑡 ∈ (𝑌𝐿𝑊) ∨ 𝑌 = 𝑊))
34	1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 22, 23, 25, 27, 29, 30, 33	symquadlem 29166	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 = (((pInvG‘𝐺)‘𝑡)‘𝑍))
35	1, 4, 3, 6, 11, 13, 9, 17	ncoltgdim2 29033	. . . . . . 7 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
36	35	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝐺Dim_TarskiG≥2)
37	1, 2, 3, 7, 36, 14, 10, 5, 22	ismidb 29288	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋 = (((pInvG‘𝐺)‘𝑡)‘𝑍) ↔ (𝑍(midG‘𝐺)𝑋) = 𝑡))
38	34, 37	mpbid 235	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑍(midG‘𝐺)𝑋) = 𝑡)
39	1, 2, 3, 7, 36, 10, 14	midcom 29292	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋(midG‘𝐺)𝑍) = (𝑍(midG‘𝐺)𝑋))
40	1, 2, 3, 7, 36, 12, 16	midcom 29292	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = (𝑊(midG‘𝐺)𝑌))
41	28	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 − 𝑋) = (𝑌 − 𝑍))
42	1, 2, 3, 6, 15, 9, 11, 13, 41	tgcgrcomlr 28947	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 − 𝑊) = (𝑍 − 𝑌))
43	42	eqcomd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 − 𝑌) = (𝑋 − 𝑊))
44	43	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑍 − 𝑌) = (𝑋 − 𝑊))
45	1, 2, 3, 6, 9, 11, 13, 15, 26	tgcgrcomlr 28947	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 − 𝑋) = (𝑊 − 𝑍))
46	45	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌 − 𝑋) = (𝑊 − 𝑍))
47	1, 4, 3, 6, 11, 13, 9, 17	ncolrot2 29031	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
48	1, 4, 3, 6, 9, 11, 13, 47	ncolcom 29029	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
49	48	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
50	1, 3, 4, 6, 9, 13, 18	tglinecom 29108	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
51	50	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
52	21, 51	eleqtrd 2862	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑍𝐿𝑋))
53	1, 2, 4, 7, 14, 12, 10, 16, 44, 46, 49, 25, 52, 32	symquadprlnglem 29170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
54	1, 4, 3, 7, 10, 12, 16, 53	ncolcom 29029	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
55	1, 4, 3, 7, 12, 10, 16, 54	ncolrot1 29030	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑌 ∈ (𝑋𝐿𝑊) ∨ 𝑋 = 𝑊))
56	18	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 ≠ 𝑍)
57	42	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋 − 𝑊) = (𝑍 − 𝑌))
58	1, 2, 3, 4, 5, 7, 8, 12, 10, 16, 14, 22, 55, 56, 46, 57, 33, 30	symquadlem 29166	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 = (((pInvG‘𝐺)‘𝑡)‘𝑊))
59	1, 2, 3, 7, 36, 16, 12, 5, 22	ismidb 29288	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌 = (((pInvG‘𝐺)‘𝑡)‘𝑊) ↔ (𝑊(midG‘𝐺)𝑌) = 𝑡))
60	58, 59	mpbid 235	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑊(midG‘𝐺)𝑌) = 𝑡)
61	40, 60	eqtrd 2795	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = 𝑡)
62	38, 39, 61	3eqtr4d 2805	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑊))
63		symquadmid.m	. . . 4 ⊢ 𝑀 = (midG‘𝐺)
64	63	oveqi 7433	. . 3 ⊢ (𝑋𝑀𝑍) = (𝑋(midG‘𝐺)𝑍)
65	63	oveqi 7433	. . 3 ⊢ (𝑌𝑀𝑊) = (𝑌(midG‘𝐺)𝑊)
66	62, 64, 65	3eqtr4g 2820	. 2 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))
67		symquadmid.6	. . . 4 ⊢ (𝜑 → 𝑌𝑂𝑊)
68		symquadmid.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}
69	1, 2, 3, 68, 11, 15	islnopp 29220	. . . 4 ⊢ (𝜑 → (𝑌𝑂𝑊 ↔ ((¬ 𝑌 ∈ (𝑋𝐿𝑍) ∧ ¬ 𝑊 ∈ (𝑋𝐿𝑍)) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑌𝐼𝑊))))
70	67, 69	mpbid 235	. . 3 ⊢ (𝜑 → ((¬ 𝑌 ∈ (𝑋𝐿𝑍) ∧ ¬ 𝑊 ∈ (𝑋𝐿𝑍)) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑌𝐼𝑊)))
71	70	simprd 501	. 2 ⊢ (𝜑 → ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑌𝐼𝑊))
72	66, 71	r19.29a 3170	1 ⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∃wrex 3086 ∖ cdif 4045 class class class wbr 5102 {copab 5166 ran crn 5651 ‘cfv 6538 (class class class)co 7420 2c2 12400 Basecbs 17392 distcds 17442 TarskiGcstrkg 28894 Dim_TarskiG≥cstrkgld 28898 Itvcitv 28900 LineGclng 28901 pInvGcmir 29129 midGcmid 29282

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-oadd 8483 df-er 8720 df-map 8852 df-pm 8853 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-dju 9985 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-n0 12610 df-xnn0 12683 df-z 12697 df-uz 12969 df-fz 13643 df-fzo 13792 df-hash 14478 df-word 14662 df-concat 14719 df-s1 14746 df-s2 15003 df-s3 15004 df-trkgc 28915 df-trkgb 28916 df-trkgcb 28917 df-trkgld 28919 df-trkg 28920 df-cgrg 28979 df-leg 29051 df-mir 29130 df-rag 29174 df-perpg 29176 df-mid 29284

This theorem is used by: (None)

W3C validator