chordthmlem4 - Metamath Proof Explorer

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Theorem chordthmlem4 25130

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		1red 10439	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
2		unitssre 12700	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
3		chordthmlem4.X	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
4	2, 3	sseldi 3851	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5	1, 4	resubcld 10868	. . . . . . 7 ⊢ (𝜑 → (1 − 𝑋) ∈ ℝ)
6	5	recnd 10467	. . . . . 6 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
7	6	abscld 14656	. . . . 5 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℝ)
8	7	recnd 10467	. . . 4 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℂ)
9		chordthmlem4.B	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10		chordthmlem4.A	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	9, 10	subcld 10797	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
12	11	abscld 14656	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℝ)
13	12	recnd 10467	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℂ)
14	4	recnd 10467	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
15	14	abscld 14656	. . . . 5 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
16	15	recnd 10467	. . . 4 ⊢ (𝜑 → (abs‘𝑋) ∈ ℂ)
17	8, 13, 16, 13	mul4d 10651	. . 3 ⊢ (𝜑 → (((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))) · ((abs‘𝑋) · (abs‘(𝐵 − 𝐴)))) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴)))))
18		chordthmlem4.P	. . . . . . 7 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
19	14, 10	mulcld 10459	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
20	6, 9	mulcld 10459	. . . . . . . . . 10 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
21	19, 20	addcld 10458	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
22	18, 21	eqeltrd 2861	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
23	10, 22, 9, 14	affineequiv2 25119	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴))))
24	18, 23	mpbid 224	. . . . . 6 ⊢ (𝜑 → (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴)))
25	24	fveq2d 6501	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = (abs‘((1 − 𝑋) · (𝐵 − 𝐴))))
26	6, 11	absmuld 14674	. . . . 5 ⊢ (𝜑 → (abs‘((1 − 𝑋) · (𝐵 − 𝐴))) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
27	25, 26	eqtrd 2809	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
28	22, 9	abssubd 14673	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = (abs‘(𝐵 − 𝑃)))
29	10, 22, 9, 14	affineequiv 25118	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
30	18, 29	mpbid 224	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))
31	30	fveq2d 6501	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝑃)) = (abs‘(𝑋 · (𝐵 − 𝐴))))
32	14, 11	absmuld 14674	. . . . 5 ⊢ (𝜑 → (abs‘(𝑋 · (𝐵 − 𝐴))) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
33	28, 31, 32	3eqtrd 2813	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
34	27, 33	oveq12d 6993	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))) · ((abs‘𝑋) · (abs‘(𝐵 − 𝐴)))))
35	13	sqvald 13321	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) = ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴))))
36	35	oveq2d 6991	. . 3 ⊢ (𝜑 → (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴)))))
37	17, 34, 36	3eqtr4d 2819	. 2 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
38	1	recnd 10467	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
39	38	halfcld 11691	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
40	39	sqcld 13322	. . . 4 ⊢ (𝜑 → ((1 / 2)↑2) ∈ ℂ)
41	1	rehalfcld 11693	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
42	41, 4	resubcld 10868	. . . . . . . 8 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
43	42	recnd 10467	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
44	43	abscld 14656	. . . . . 6 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℝ)
45	44	recnd 10467	. . . . 5 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℂ)
46	45	sqcld 13322	. . . 4 ⊢ (𝜑 → ((abs‘((1 / 2) − 𝑋))↑2) ∈ ℂ)
47	13	sqcld 13322	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) ∈ ℂ)
48	40, 46, 47	subdird 10897	. . 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))))
49		subsq 13386	. . . . . . 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))))
50	39, 43, 49	syl2anc 576	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)) = (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))))
51	39, 39, 14	addsubassd 10817	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = ((1 / 2) + ((1 / 2) − 𝑋)))
52	38	2halvesd 11692	. . . . . . . . 9 ⊢ (𝜑 → ((1 / 2) + (1 / 2)) = 1)
53	52	oveq1d 6990	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = (1 − 𝑋))
54	51, 53	eqtr3d 2811	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) + ((1 / 2) − 𝑋)) = (1 − 𝑋))
55	39, 14	nncand 10802	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − ((1 / 2) − 𝑋)) = 𝑋)
56	54, 55	oveq12d 6993	. . . . . 6 ⊢ (𝜑 → (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))) = ((1 − 𝑋) · 𝑋))
57	50, 56	eqtr2d 2810	. . . . 5 ⊢ (𝜑 → ((1 − 𝑋) · 𝑋) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
58		elicc01 12669	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
59	3, 58	sylib 210	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
60	59	simp3d 1125	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 1)
61	4, 1, 60	abssubge0d 14651	. . . . . 6 ⊢ (𝜑 → (abs‘(1 − 𝑋)) = (1 − 𝑋))
62	59	simp2d 1124	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑋)
63	4, 62	absidd 14642	. . . . . 6 ⊢ (𝜑 → (abs‘𝑋) = 𝑋)
64	61, 63	oveq12d 6993	. . . . 5 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = ((1 − 𝑋) · 𝑋))
65		absresq 14522	. . . . . . 7 ⊢ (((1 / 2) − 𝑋) ∈ ℝ → ((abs‘((1 / 2) − 𝑋))↑2) = (((1 / 2) − 𝑋)↑2))
66	42, 65	syl 17	. . . . . 6 ⊢ (𝜑 → ((abs‘((1 / 2) − 𝑋))↑2) = (((1 / 2) − 𝑋)↑2))
67	66	oveq2d 6991	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
68	57, 64, 67	3eqtr4d 2819	. . . 4 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)))
69	68	oveq1d 6990	. . 3 ⊢ (𝜑 → (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)) = ((((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)) · ((abs‘(𝐵 − 𝐴))↑2)))
70		2cnd 11517	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
71		2ne0 11550	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
72	71	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
73	9, 70, 72	divcan4d 11222	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
74	9	times2d 11690	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
75	74	oveq1d 6990	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
76	73, 75	eqtr3d 2811	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
77		chordthmlem4.M	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
78	76, 77	oveq12d 6993	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
79	9, 9	addcld 10458	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
80	10, 9	addcld 10458	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
81	79, 80, 70, 72	divsubdird 11255	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
82	9, 10, 9	pnpcan2d 10835	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
83	82	oveq1d 6990	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
84	78, 81, 83	3eqtr2d 2815	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
85	11, 70, 72	divrec2d 11220	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
86	84, 85	eqtrd 2809	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
87	86	fveq2d 6501	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = (abs‘((1 / 2) · (𝐵 − 𝐴))))
88	39, 11	absmuld 14674	. . . . . . 7 ⊢ (𝜑 → (abs‘((1 / 2) · (𝐵 − 𝐴))) = ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))))
89		0red 10442	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
90		halfgt0 11662	. . . . . . . . . . 11 ⊢ 0 < (1 / 2)
91	90	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 / 2))
92	89, 41, 91	ltled 10587	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (1 / 2))
93	41, 92	absidd 14642	. . . . . . . 8 ⊢ (𝜑 → (abs‘(1 / 2)) = (1 / 2))
94	93	oveq1d 6990	. . . . . . 7 ⊢ (𝜑 → ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
95	87, 88, 94	3eqtrd 2813	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
96	95	oveq1d 6990	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2))
97	39, 13	sqmuld 13336	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
98	96, 97	eqtrd 2809	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
99	39, 14, 11	subdird 10897	. . . . . . . . 9 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
100	86, 30	oveq12d 6993	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
101	80	halfcld 11691	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
102	77, 101	eqeltrd 2861	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
103	9, 102, 22	nnncan1d 10831	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
104	99, 100, 103	3eqtr2rd 2816	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
105	104	fveq2d 6501	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
106	43, 11	absmuld 14674	. . . . . . 7 ⊢ (𝜑 → (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
107	105, 106	eqtrd 2809	. . . . . 6 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
108	107	oveq1d 6990	. . . . 5 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2))
109	45, 13	sqmuld 13336	. . . . 5 ⊢ (𝜑 → (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
110	108, 109	eqtrd 2809	. . . 4 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
111	98, 110	oveq12d 6993	. . 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))))
112	48, 69, 111	3eqtr4rd 2820	. 2 ⊢ (𝜑 → (((abs‘(𝐵 − 𝑀))↑2) − ((abs‘(𝑃 − 𝑀))↑2)) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
113	37, 112	eqtr4d 2812	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 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 ℂcc 10332 ℝcr 10333 0cc0 10334 1c1 10335 + caddc 10337 · cmul 10339 < clt 10473 ≤ cle 10474 − cmin 10669 / cdiv 11097 2c2 11494 [,]cicc 12556 ↑cexp 13243 abscabs 14453

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-z 11793 df-uz 12058 df-rp 12204 df-icc 12560 df-seq 13184 df-exp 13244 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455

This theorem is referenced by: chordthmlem5 25131

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