Theorem symquadmid 29286

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 2761	. . . . . 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 2761	. . . . . 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 29076	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
19	1, 3, 4, 6, 9, 13, 18	tgelrnln 29080	. . . . . . . 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 28994	. . . . . 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 29069	. . . . . . 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 29143	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 = (((pInvG‘𝐺)‘𝑡)‘𝑍))
35	1, 4, 3, 6, 11, 13, 9, 17	ncoltgdim2 29010	. . . . . . 7 ⊢ (𝜑 → 𝐺DimTarskiG≥2)
36	35	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝐺DimTarskiG≥2)
37	1, 2, 3, 7, 36, 14, 10, 5, 22	ismidb 29265	. . . . 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 29269	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋(midG‘𝐺)𝑍) = (𝑍(midG‘𝐺)𝑋))
40	1, 2, 3, 7, 36, 12, 16	midcom 29269	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = (𝑊(midG‘𝐺)𝑌))
41	28	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 − 𝑋) = (𝑌 − 𝑍))
42	1, 2, 3, 6, 15, 9, 11, 13, 41	tgcgrcomlr 28924	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 − 𝑊) = (𝑍 − 𝑌))
43	42	eqcomd 2767	. . . . . . . . . . 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 28924	. . . . . . . . . . 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 29008	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
48	1, 4, 3, 6, 9, 11, 13, 47	ncolcom 29006	. . . . . . . . . . 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 29085	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
51	50	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
52	21, 51	eleqtrd 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑍𝐿𝑋))
53	1, 2, 4, 7, 14, 12, 10, 16, 44, 46, 49, 25, 52, 32	symquadprlnglem 29147	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
54	1, 4, 3, 7, 10, 12, 16, 53	ncolcom 29006	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
55	1, 4, 3, 7, 12, 10, 16, 54	ncolrot1 29007	. . . . . . 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 29143	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 = (((pInvG‘𝐺)‘𝑡)‘𝑊))
59	1, 2, 3, 7, 36, 16, 12, 5, 22	ismidb 29265	. . . . . 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 2796	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = 𝑡)
62	38, 39, 61	3eqtr4d 2806	. . 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 2821	. 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 29197	. . . 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 3171	1 ⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ran crn 5652  ‘cfv 6531  (class class class)co 7412  2c2 12378  Basecbs 17367  distcds 17417  TarskiGcstrkg 28871  Dim_TarskiG≥cstrkgld 28875  Itvcitv 28877  LineGclng 28878  pInvGcmir 29106  midGcmid 29259

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-oadd 8464  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9963  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-xnn0 12661  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-hash 14455  df-word 14639  df-concat 14696  df-s1 14723  df-s2 14979  df-s3 14980  df-trkgc 28892  df-trkgb 28893  df-trkgcb 28894  df-trkgld 28896  df-trkg 28897  df-cgrg 28956  df-leg 29028  df-mir 29107  df-rag 29151  df-perpg 29153  df-mid 29261

This theorem is used by: (None)