chordthmlem4 - Metamath Proof Explorer

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Theorem chordthmlem4 26813

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 11145	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
2		unitssre 13427	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
3		chordthmlem4.X	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
4	2, 3	sselid 3933	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5	1, 4	resubcld 11577	. . . . . . 7 ⊢ (𝜑 → (1 − 𝑋) ∈ ℝ)
6	5	recnd 11172	. . . . . 6 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
7	6	abscld 15374	. . . . 5 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℝ)
8	7	recnd 11172	. . . 4 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℂ)
9		chordthmlem4.B	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10		chordthmlem4.A	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	9, 10	subcld 11504	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
12	11	abscld 15374	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℝ)
13	12	recnd 11172	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℂ)
14	4	recnd 11172	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
15	14	abscld 15374	. . . . 5 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
16	15	recnd 11172	. . . 4 ⊢ (𝜑 → (abs‘𝑋) ∈ ℂ)
17	8, 13, 16, 13	mul4d 11357	. . 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 11164	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
20	6, 9	mulcld 11164	. . . . . . . . . 10 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
21	19, 20	addcld 11163	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
22	18, 21	eqeltrd 2837	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
23	10, 22, 9, 14	affineequiv2 26802	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴))))
24	18, 23	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴)))
25	24	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = (abs‘((1 − 𝑋) · (𝐵 − 𝐴))))
26	6, 11	absmuld 15392	. . . . 5 ⊢ (𝜑 → (abs‘((1 − 𝑋) · (𝐵 − 𝐴))) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
27	25, 26	eqtrd 2772	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
28	22, 9	abssubd 15391	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = (abs‘(𝐵 − 𝑃)))
29	10, 22, 9, 14	affineequiv 26801	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
30	18, 29	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))
31	30	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝑃)) = (abs‘(𝑋 · (𝐵 − 𝐴))))
32	14, 11	absmuld 15392	. . . . 5 ⊢ (𝜑 → (abs‘(𝑋 · (𝐵 − 𝐴))) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
33	28, 31, 32	3eqtrd 2776	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
34	27, 33	oveq12d 7386	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))) · ((abs‘𝑋) · (abs‘(𝐵 − 𝐴)))))
35	13	sqvald 14078	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) = ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴))))
36	35	oveq2d 7384	. . 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 2782	. 2 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
38	1	recnd 11172	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
39	38	halfcld 12398	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
40	39	sqcld 14079	. . . 4 ⊢ (𝜑 → ((1 / 2)↑2) ∈ ℂ)
41	1	rehalfcld 12400	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
42	41, 4	resubcld 11577	. . . . . . . 8 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
43	42	recnd 11172	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
44	43	abscld 15374	. . . . . 6 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℝ)
45	44	recnd 11172	. . . . 5 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℂ)
46	45	sqcld 14079	. . . 4 ⊢ (𝜑 → ((abs‘((1 / 2) − 𝑋))↑2) ∈ ℂ)
47	13	sqcld 14079	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) ∈ ℂ)
48	40, 46, 47	subdird 11606	. . 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 14145	. . . . . . 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 585	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)) = (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))))
51	39, 39, 14	addsubassd 11524	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = ((1 / 2) + ((1 / 2) − 𝑋)))
52	38	2halvesd 12399	. . . . . . . . 9 ⊢ (𝜑 → ((1 / 2) + (1 / 2)) = 1)
53	52	oveq1d 7383	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = (1 − 𝑋))
54	51, 53	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) + ((1 / 2) − 𝑋)) = (1 − 𝑋))
55	39, 14	nncand 11509	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − ((1 / 2) − 𝑋)) = 𝑋)
56	54, 55	oveq12d 7386	. . . . . 6 ⊢ (𝜑 → (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))) = ((1 − 𝑋) · 𝑋))
57	50, 56	eqtr2d 2773	. . . . 5 ⊢ (𝜑 → ((1 − 𝑋) · 𝑋) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
58		elicc01 13394	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
59	3, 58	sylib 218	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
60	59	simp3d 1145	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 1)
61	4, 1, 60	abssubge0d 15369	. . . . . 6 ⊢ (𝜑 → (abs‘(1 − 𝑋)) = (1 − 𝑋))
62	59	simp2d 1144	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑋)
63	4, 62	absidd 15358	. . . . . 6 ⊢ (𝜑 → (abs‘𝑋) = 𝑋)
64	61, 63	oveq12d 7386	. . . . 5 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = ((1 − 𝑋) · 𝑋))
65		absresq 15237	. . . . . . 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 7384	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
68	57, 64, 67	3eqtr4d 2782	. . . 4 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)))
69	68	oveq1d 7383	. . 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 12235	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
71		2ne0 12261	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
72	71	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
73	9, 70, 72	divcan4d 11935	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
74	9	times2d 12397	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
75	74	oveq1d 7383	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
76	73, 75	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
77		chordthmlem4.M	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
78	76, 77	oveq12d 7386	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
79	9, 9	addcld 11163	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
80	10, 9	addcld 11163	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
81	79, 80, 70, 72	divsubdird 11968	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
82	9, 10, 9	pnpcan2d 11542	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
83	82	oveq1d 7383	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
84	78, 81, 83	3eqtr2d 2778	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
85	11, 70, 72	divrec2d 11933	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
86	84, 85	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
87	86	fveq2d 6846	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = (abs‘((1 / 2) · (𝐵 − 𝐴))))
88	39, 11	absmuld 15392	. . . . . . 7 ⊢ (𝜑 → (abs‘((1 / 2) · (𝐵 − 𝐴))) = ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))))
89		0red 11147	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
90		halfgt0 12368	. . . . . . . . . . 11 ⊢ 0 < (1 / 2)
91	90	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 / 2))
92	89, 41, 91	ltled 11293	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (1 / 2))
93	41, 92	absidd 15358	. . . . . . . 8 ⊢ (𝜑 → (abs‘(1 / 2)) = (1 / 2))
94	93	oveq1d 7383	. . . . . . 7 ⊢ (𝜑 → ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
95	87, 88, 94	3eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
96	95	oveq1d 7383	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2))
97	39, 13	sqmuld 14093	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
98	96, 97	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
99	39, 14, 11	subdird 11606	. . . . . . . . 9 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
100	86, 30	oveq12d 7386	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
101	80	halfcld 12398	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
102	77, 101	eqeltrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
103	9, 102, 22	nnncan1d 11538	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
104	99, 100, 103	3eqtr2rd 2779	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
105	104	fveq2d 6846	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
106	43, 11	absmuld 15392	. . . . . . 7 ⊢ (𝜑 → (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
107	105, 106	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
108	107	oveq1d 7383	. . . . 5 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2))
109	45, 13	sqmuld 14093	. . . . 5 ⊢ (𝜑 → (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
110	108, 109	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
111	98, 110	oveq12d 7386	. . 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 2783	. 2 ⊢ (𝜑 → (((abs‘(𝐵 − 𝑀))↑2) − ((abs‘(𝑃 − 𝑀))↑2)) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
113	37, 112	eqtr4d 2775	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 ≤ cle 11179 − cmin 11376 / cdiv 11806 2c2 12212 [,]cicc 13276 ↑cexp 13996 abscabs 15169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-icc 13280 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171

This theorem is referenced by: chordthmlem5 26814

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