chordthmlem4 - Metamath Proof Explorer

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Theorem chordthmlem4 24562

Description: If P is on the segment AB and M is the midpoint of AB, then PA · PB = BM ² − PM ² . If all lengths are reexpressed as fractions of AB, this reduces to the identity 𝑋 · (1 − 𝑋) = (1 / 2) ² − ((1 / 2) − 𝑋) ² . (Contributed by David Moews, 28-Feb-2017.)

**Hypotheses**
Ref	Expression
chordthmlem4.A	⊢ (𝜑 → 𝐴 ∈ ℂ)
chordthmlem4.B	⊢ (𝜑 → 𝐵 ∈ ℂ)
chordthmlem4.X	⊢ (𝜑 → 𝑋 ∈ (0[,]1))
chordthmlem4.M	⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
chordthmlem4.P	⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))

**Assertion**
Ref	Expression
chordthmlem4	⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − 𝑀))↑2) − ((abs‘(𝑃 − 𝑀))↑2)))

**Proof of Theorem chordthmlem4**
Step	Hyp	Ref	Expression
1		1re 10039	. . . . . . . . 9 ⊢ 1 ∈ ℝ
2	1	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
3		unitssre 12319	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
4		chordthmlem4.X	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
5	3, 4	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
6	2, 5	resubcld 10458	. . . . . . 7 ⊢ (𝜑 → (1 − 𝑋) ∈ ℝ)
7	6	recnd 10068	. . . . . 6 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
8	7	abscld 14175	. . . . 5 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℝ)
9	8	recnd 10068	. . . 4 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℂ)
10		chordthmlem4.B	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
11		chordthmlem4.A	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10, 11	subcld 10392	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
13	12	abscld 14175	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℝ)
14	13	recnd 10068	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℂ)
15	5	recnd 10068	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
16	15	abscld 14175	. . . . 5 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
17	16	recnd 10068	. . . 4 ⊢ (𝜑 → (abs‘𝑋) ∈ ℂ)
18	9, 14, 17, 14	mul4d 10248	. . 3 ⊢ (𝜑 → (((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))) · ((abs‘𝑋) · (abs‘(𝐵 − 𝐴)))) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴)))))
19		chordthmlem4.P	. . . . . . 7 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
20	15, 11	mulcld 10060	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
21	7, 10	mulcld 10060	. . . . . . . . . 10 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
22	20, 21	addcld 10059	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
23	19, 22	eqeltrd 2701	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
24	11, 23, 10, 15	affineequiv2 24554	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴))))
25	19, 24	mpbid 222	. . . . . 6 ⊢ (𝜑 → (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴)))
26	25	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = (abs‘((1 − 𝑋) · (𝐵 − 𝐴))))
27	7, 12	absmuld 14193	. . . . 5 ⊢ (𝜑 → (abs‘((1 − 𝑋) · (𝐵 − 𝐴))) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
28	26, 27	eqtrd 2656	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
29	23, 10	abssubd 14192	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = (abs‘(𝐵 − 𝑃)))
30	11, 23, 10, 15	affineequiv 24553	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
31	19, 30	mpbid 222	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))
32	31	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝑃)) = (abs‘(𝑋 · (𝐵 − 𝐴))))
33	15, 12	absmuld 14193	. . . . 5 ⊢ (𝜑 → (abs‘(𝑋 · (𝐵 − 𝐴))) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
34	29, 32, 33	3eqtrd 2660	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
35	28, 34	oveq12d 6668	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))) · ((abs‘𝑋) · (abs‘(𝐵 − 𝐴)))))
36	14	sqvald 13005	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) = ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴))))
37	36	oveq2d 6666	. . 3 ⊢ (𝜑 → (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴)))))
38	18, 35, 37	3eqtr4d 2666	. 2 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
39	2	recnd 10068	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
40	39	halfcld 11277	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
41	40	sqcld 13006	. . . 4 ⊢ (𝜑 → ((1 / 2)↑2) ∈ ℂ)
42	2	rehalfcld 11279	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
43	42, 5	resubcld 10458	. . . . . . . 8 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
44	43	recnd 10068	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
45	44	abscld 14175	. . . . . 6 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℝ)
46	45	recnd 10068	. . . . 5 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℂ)
47	46	sqcld 13006	. . . 4 ⊢ (𝜑 → ((abs‘((1 / 2) − 𝑋))↑2) ∈ ℂ)
48	14	sqcld 13006	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) ∈ ℂ)
49	41, 47, 48	subdird 10487	. . 3 ⊢ (𝜑 → ((((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)) · ((abs‘(𝐵 − 𝐴))↑2)) = ((((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)) − (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2))))
50		subsq 12972	. . . . . . 7 ⊢ (((1 / 2) ∈ ℂ ∧ ((1 / 2) − 𝑋) ∈ ℂ) → (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)) = (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))))
51	40, 44, 50	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)) = (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))))
52	40, 40, 15	addsubassd 10412	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = ((1 / 2) + ((1 / 2) − 𝑋)))
53	39	2halvesd 11278	. . . . . . . . 9 ⊢ (𝜑 → ((1 / 2) + (1 / 2)) = 1)
54	53	oveq1d 6665	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = (1 − 𝑋))
55	52, 54	eqtr3d 2658	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) + ((1 / 2) − 𝑋)) = (1 − 𝑋))
56	40, 15	nncand 10397	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − ((1 / 2) − 𝑋)) = 𝑋)
57	55, 56	oveq12d 6668	. . . . . 6 ⊢ (𝜑 → (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))) = ((1 − 𝑋) · 𝑋))
58	51, 57	eqtr2d 2657	. . . . 5 ⊢ (𝜑 → ((1 − 𝑋) · 𝑋) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
59		0re 10040	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
60	59, 1	elicc2i 12239	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
61	4, 60	sylib 208	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
62	61	simp3d 1075	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 1)
63	5, 2, 62	abssubge0d 14170	. . . . . 6 ⊢ (𝜑 → (abs‘(1 − 𝑋)) = (1 − 𝑋))
64	61	simp2d 1074	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑋)
65	5, 64	absidd 14161	. . . . . 6 ⊢ (𝜑 → (abs‘𝑋) = 𝑋)
66	63, 65	oveq12d 6668	. . . . 5 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = ((1 − 𝑋) · 𝑋))
67		absresq 14042	. . . . . . 7 ⊢ (((1 / 2) − 𝑋) ∈ ℝ → ((abs‘((1 / 2) − 𝑋))↑2) = (((1 / 2) − 𝑋)↑2))
68	43, 67	syl 17	. . . . . 6 ⊢ (𝜑 → ((abs‘((1 / 2) − 𝑋))↑2) = (((1 / 2) − 𝑋)↑2))
69	68	oveq2d 6666	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
70	58, 66, 69	3eqtr4d 2666	. . . 4 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)))
71	70	oveq1d 6665	. . 3 ⊢ (𝜑 → (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)) = ((((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)) · ((abs‘(𝐵 − 𝐴))↑2)))
72		2cnd 11093	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
73		2ne0 11113	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
74	73	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
75	10, 72, 74	divcan4d 10807	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
76	10	times2d 11276	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
77	76	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
78	75, 77	eqtr3d 2658	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
79		chordthmlem4.M	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
80	78, 79	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
81	10, 10	addcld 10059	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
82	11, 10	addcld 10059	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
83	81, 82, 72, 74	divsubdird 10840	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
84	10, 11, 10	pnpcan2d 10430	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
85	84	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
86	80, 83, 85	3eqtr2d 2662	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
87	12, 72, 74	divrec2d 10805	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
88	86, 87	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
89	88	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = (abs‘((1 / 2) · (𝐵 − 𝐴))))
90	40, 12	absmuld 14193	. . . . . . 7 ⊢ (𝜑 → (abs‘((1 / 2) · (𝐵 − 𝐴))) = ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))))
91	59	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
92		halfgt0 11248	. . . . . . . . . . 11 ⊢ 0 < (1 / 2)
93	92	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 / 2))
94	91, 42, 93	ltled 10185	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (1 / 2))
95	42, 94	absidd 14161	. . . . . . . 8 ⊢ (𝜑 → (abs‘(1 / 2)) = (1 / 2))
96	95	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
97	89, 90, 96	3eqtrd 2660	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
98	97	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2))
99	40, 14	sqmuld 13020	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
100	98, 99	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
101	40, 15, 12	subdird 10487	. . . . . . . . 9 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
102	88, 31	oveq12d 6668	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
103	82	halfcld 11277	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
104	79, 103	eqeltrd 2701	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
105	10, 104, 23	nnncan1d 10426	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
106	101, 102, 105	3eqtr2rd 2663	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
107	106	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
108	44, 12	absmuld 14193	. . . . . . 7 ⊢ (𝜑 → (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
109	107, 108	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
110	109	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2))
111	46, 14	sqmuld 13020	. . . . 5 ⊢ (𝜑 → (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
112	110, 111	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
113	100, 112	oveq12d 6668	. . 3 ⊢ (𝜑 → (((abs‘(𝐵 − 𝑀))↑2) − ((abs‘(𝑃 − 𝑀))↑2)) = ((((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)) − (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2))))
114	49, 71, 113	3eqtr4rd 2667	. 2 ⊢ (𝜑 → (((abs‘(𝐵 − 𝑀))↑2) − ((abs‘(𝑃 − 𝑀))↑2)) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
115	38, 114	eqtr4d 2659	1 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − 𝑀))↑2) − ((abs‘(𝑃 − 𝑀))↑2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 2c2 11070 [,]cicc 12178 ↑cexp 12860 abscabs 13974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-icc 12182 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: chordthmlem5 24563

W3C validator