symquadmid - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > symquadmid		Structured version Visualization version GIF version

Theorem symquadmid 29145

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 2766	. . . . . 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 2766	. . . . . 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 28936	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
19	1, 3, 4, 6, 9, 13, 18	tgelrnln 28940	. . . . . . . 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 28855	. . . . . 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 28929	. . . . . . 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 29003	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑋 = (((pInvG‘𝐺)‘𝑡)‘𝑍))
35	1, 4, 3, 6, 11, 13, 9, 17	ncoltgdim2 28871	. . . . . . 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 29124	. . . . 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 29128	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋(midG‘𝐺)𝑍) = (𝑍(midG‘𝐺)𝑋))
40	1, 2, 3, 7, 36, 12, 16	midcom 29128	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = (𝑊(midG‘𝐺)𝑌))
41	28	eqcomd 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 − 𝑋) = (𝑌 − 𝑍))
42	1, 2, 3, 6, 15, 9, 11, 13, 41	tgcgrcomlr 28786	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 − 𝑊) = (𝑍 − 𝑌))
43	42	eqcomd 2772	. . . . . . . . . . 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 28786	. . . . . . . . . . 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 28869	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
48	1, 4, 3, 6, 9, 11, 13, 47	ncolcom 28867	. . . . . . . . . . 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 28945	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
51	50	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
52	21, 51	eleqtrd 2868	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑡 ∈ (𝑍𝐿𝑋))
53	1, 2, 4, 7, 14, 12, 10, 16, 44, 46, 49, 25, 52, 32	symquadprlnglem 29007	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
54	1, 4, 3, 7, 10, 12, 16, 53	ncolcom 28867	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → ¬ (𝑊 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
55	1, 4, 3, 7, 12, 10, 16, 54	ncolrot1 28868	. . . . . . 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 29003	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → 𝑌 = (((pInvG‘𝐺)‘𝑡)‘𝑊))
59	1, 2, 3, 7, 36, 16, 12, 5, 22	ismidb 29124	. . . . . 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 2801	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑌(midG‘𝐺)𝑊) = 𝑡)
62	38, 39, 61	3eqtr4d 2811	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑋𝐿𝑍)) ∧ 𝑡 ∈ (𝑌𝐼𝑊)) → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑊))
63		symquadmid.m	. . . 4 ⊢ 𝑀 = (midG‘𝐺)
64	63	oveqi 7436	. . 3 ⊢ (𝑋𝑀𝑍) = (𝑋(midG‘𝐺)𝑍)
65	63	oveqi 7436	. . 3 ⊢ (𝑌𝑀𝑊) = (𝑌(midG‘𝐺)𝑊)
66	62, 64, 65	3eqtr4g 2826	. 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 29057	. . . 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 3176	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 1570 ∈ wcel 2146 ≠ wne 2961 ∃wrex 3092 ∖ cdif 3905 class class class wbr 5114 {copab 5178 ran crn 5667 ‘cfv 6543 (class class class)co 7423 2c2 12313 Basecbs 17294 distcds 17344 TarskiGcstrkg 28733 Dim_TarskiG≥cstrkgld 28737 Itvcitv 28739 LineGclng 28740 pInvGcmir 28966 midGcmid 29118

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-fz 13554 df-fzo 13702 df-hash 14387 df-word 14571 df-concat 14628 df-s1 14655 df-s2 14911 df-s3 14912 df-trkgc 28754 df-trkgb 28755 df-trkgcb 28756 df-trkgld 28758 df-trkg 28759 df-cgrg 28817 df-leg 28889 df-mir 28967 df-rag 29011 df-perpg 29013 df-mid 29120

This theorem is used by: (None)

W3C validator