Theorem symquadmid 29086

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 2763	. . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		symquadmid.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝐺 ∈ TarskiG)
8		eqid 2763	. . . . . 6 ⊢ ((pInvG‘𝐺)‘𝑡) = ((pInvG‘𝐺)‘𝑡)
9		symquadmid.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 ∈ 𝑃)
11		symquadmid.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
12	11	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 ∈ 𝑃)
13		symquadmid.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
14	13	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑍 ∈ 𝑃)
15		symquadmid.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
16	15	ad2antrr 738	. . . . . 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 28877	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
19	1, 3, 4, 6, 9, 13, 18	tgelrnln 28881	. . . . . . . 8 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
20	19	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋𝐿𝑍) ∈ ran 𝐿)
21		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑋𝐿𝑍))
22	1, 4, 3, 7, 20, 21	tglnpt 28796	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ 𝑃)
23	17	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
24		symquadmid.3	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
25	24	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 ≠ 𝑊)
26		symquadmid.4	. . . . . . 7 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
27	26	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋 − 𝑌) = (𝑍 − 𝑊))
28		symquadmid.5	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
29	28	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌 − 𝑍) = (𝑊 − 𝑋))
30	21	orcd 886	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑡 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
31		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑌𝐼𝑊))
32	1, 3, 4, 7, 12, 16, 22, 25, 31	btwnlng1 28870	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑌𝐿𝑊))
33	32	orcd 886	. . . . . 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 28944	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 = (((pInvG‘𝐺)‘𝑡)‘𝑍))
35	1, 4, 3, 6, 11, 13, 9, 17	ncoltgdim2 28812	. . . . . . 7 ⊢ (𝜑 → 𝐺DimTarskiG≥2)
36	35	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝐺DimTarskiG≥2)
37	1, 2, 3, 7, 36, 14, 10, 5, 22	ismidb 29065	. . . . 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 29069	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋(midG‘𝐺)𝑍) = (𝑍(midG‘𝐺)𝑋))
40	1, 2, 3, 7, 36, 12, 16	midcom 29069	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = (𝑊(midG‘𝐺)𝑌))
41	28	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 − 𝑋) = (𝑌 − 𝑍))
42	1, 2, 3, 6, 15, 9, 11, 13, 41	tgcgrcomlr 28727	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 − 𝑊) = (𝑍 − 𝑌))
43	42	eqcomd 2769	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 − 𝑌) = (𝑋 − 𝑊))
44	43	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑍 − 𝑌) = (𝑋 − 𝑊))
45	1, 2, 3, 6, 9, 11, 13, 15, 26	tgcgrcomlr 28727	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 − 𝑋) = (𝑊 − 𝑍))
46	45	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌 − 𝑋) = (𝑊 − 𝑍))
47	1, 4, 3, 6, 11, 13, 9, 17	ncolrot2 28810	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
48	1, 4, 3, 6, 9, 11, 13, 47	ncolcom 28808	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
49	48	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
50	1, 3, 4, 6, 9, 13, 18	tglinecom 28886	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
51	50	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
52	21, 51	eleqtrd 2865	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑍𝐿𝑋))
53	1, 2, 4, 7, 14, 12, 10, 16, 44, 46, 49, 25, 52, 32	symquadprlnglem 28948	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
54	1, 4, 3, 7, 10, 12, 16, 53	ncolcom 28808	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
55	1, 4, 3, 7, 12, 10, 16, 54	ncolrot1 28809	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑌 ∈ (𝑋𝐿𝑊) ∨ 𝑋 = 𝑊))
56	18	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 ≠ 𝑍)
57	42	ad2antrr 738	. . . . . . 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 28944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 = (((pInvG‘𝐺)‘𝑡)‘𝑊))
59	1, 2, 3, 7, 36, 16, 12, 5, 22	ismidb 29065	. . . . . 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 2798	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = 𝑡)
62	38, 39, 61	3eqtr4d 2808	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑊))
63		symquadmid.m	. . . 4 ⊢ 𝑀 = (midG‘𝐺)
64	63	oveqi 7425	. . 3 ⊢ (𝑋𝑀𝑍) = (𝑋(midG‘𝐺)𝑍)
65	63	oveqi 7425	. . 3 ⊢ (𝑌𝑀𝑊) = (𝑌(midG‘𝐺)𝑊)
66	62, 64, 65	3eqtr4g 2823	. 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 28998	. . . 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 500	. 2 ⊢ (𝜑 → ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑌𝐼𝑊))
72	66, 71	r19.29a 3173	1 ⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ∖ cdif 3903   class class class wbr 5110  {copab 5174  ran crn 5664  ‘cfv 6538  (class class class)co 7412  2c2 12296  Basecbs 17270  distcds 17320  TarskiGcstrkg 28674  Dim_TarskiG≥cstrkgld 28678  Itvcitv 28680  LineGclng 28681  pInvGcmir 28907  midGcmid 29059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-s2 14887  df-s3 14888  df-trkgc 28695  df-trkgb 28696  df-trkgcb 28697  df-trkgld 28699  df-trkg 28700  df-cgrg 28758  df-leg 28830  df-mir 28908  df-rag 28952  df-perpg 28954  df-mid 29061

This theorem is referenced by: (None)