chordthmlem4 - Metamath Proof Explorer

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Theorem chordthmlem4 26780

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 11240	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
2		unitssre 13503	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
3		chordthmlem4.X	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
4	2, 3	sselid 3971	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5	1, 4	resubcld 11667	. . . . . . 7 ⊢ (𝜑 → (1 − 𝑋) ∈ ℝ)
6	5	recnd 11267	. . . . . 6 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
7	6	abscld 15410	. . . . 5 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℝ)
8	7	recnd 11267	. . . 4 ⊢ (𝜑 → (abs‘(1 − 𝑋)) ∈ ℂ)
9		chordthmlem4.B	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10		chordthmlem4.A	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	9, 10	subcld 11596	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
12	11	abscld 15410	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℝ)
13	12	recnd 11267	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℂ)
14	4	recnd 11267	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
15	14	abscld 15410	. . . . 5 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ)
16	15	recnd 11267	. . . 4 ⊢ (𝜑 → (abs‘𝑋) ∈ ℂ)
17	8, 13, 16, 13	mul4d 11451	. . 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 11259	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
20	6, 9	mulcld 11259	. . . . . . . . . 10 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
21	19, 20	addcld 11258	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
22	18, 21	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
23	10, 22, 9, 14	affineequiv2 26769	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴))))
24	18, 23	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑃 − 𝐴) = ((1 − 𝑋) · (𝐵 − 𝐴)))
25	24	fveq2d 6894	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = (abs‘((1 − 𝑋) · (𝐵 − 𝐴))))
26	6, 11	absmuld 15428	. . . . 5 ⊢ (𝜑 → (abs‘((1 − 𝑋) · (𝐵 − 𝐴))) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
27	25, 26	eqtrd 2765	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐴)) = ((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))))
28	22, 9	abssubd 15427	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = (abs‘(𝐵 − 𝑃)))
29	10, 22, 9, 14	affineequiv 26768	. . . . . . 7 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
30	18, 29	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))
31	30	fveq2d 6894	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − 𝑃)) = (abs‘(𝑋 · (𝐵 − 𝐴))))
32	14, 11	absmuld 15428	. . . . 5 ⊢ (𝜑 → (abs‘(𝑋 · (𝐵 − 𝐴))) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
33	28, 31, 32	3eqtrd 2769	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − 𝐵)) = ((abs‘𝑋) · (abs‘(𝐵 − 𝐴))))
34	27, 33	oveq12d 7431	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘(𝐵 − 𝐴))) · ((abs‘𝑋) · (abs‘(𝐵 − 𝐴)))))
35	13	sqvald 14134	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) = ((abs‘(𝐵 − 𝐴)) · (abs‘(𝐵 − 𝐴))))
36	35	oveq2d 7429	. . 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 2775	. 2 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
38	1	recnd 11267	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
39	38	halfcld 12482	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
40	39	sqcld 14135	. . . 4 ⊢ (𝜑 → ((1 / 2)↑2) ∈ ℂ)
41	1	rehalfcld 12484	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
42	41, 4	resubcld 11667	. . . . . . . 8 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
43	42	recnd 11267	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
44	43	abscld 15410	. . . . . 6 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℝ)
45	44	recnd 11267	. . . . 5 ⊢ (𝜑 → (abs‘((1 / 2) − 𝑋)) ∈ ℂ)
46	45	sqcld 14135	. . . 4 ⊢ (𝜑 → ((abs‘((1 / 2) − 𝑋))↑2) ∈ ℂ)
47	13	sqcld 14135	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝐴))↑2) ∈ ℂ)
48	40, 46, 47	subdird 11696	. . 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 14200	. . . . . . 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 582	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)) = (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))))
51	39, 39, 14	addsubassd 11616	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = ((1 / 2) + ((1 / 2) − 𝑋)))
52	38	2halvesd 12483	. . . . . . . . 9 ⊢ (𝜑 → ((1 / 2) + (1 / 2)) = 1)
53	52	oveq1d 7428	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) + (1 / 2)) − 𝑋) = (1 − 𝑋))
54	51, 53	eqtr3d 2767	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) + ((1 / 2) − 𝑋)) = (1 − 𝑋))
55	39, 14	nncand 11601	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − ((1 / 2) − 𝑋)) = 𝑋)
56	54, 55	oveq12d 7431	. . . . . 6 ⊢ (𝜑 → (((1 / 2) + ((1 / 2) − 𝑋)) · ((1 / 2) − ((1 / 2) − 𝑋))) = ((1 − 𝑋) · 𝑋))
57	50, 56	eqtr2d 2766	. . . . 5 ⊢ (𝜑 → ((1 − 𝑋) · 𝑋) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
58		elicc01 13470	. . . . . . . . 9 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
59	3, 58	sylib 217	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
60	59	simp3d 1141	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 1)
61	4, 1, 60	abssubge0d 15405	. . . . . 6 ⊢ (𝜑 → (abs‘(1 − 𝑋)) = (1 − 𝑋))
62	59	simp2d 1140	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑋)
63	4, 62	absidd 15396	. . . . . 6 ⊢ (𝜑 → (abs‘𝑋) = 𝑋)
64	61, 63	oveq12d 7431	. . . . 5 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = ((1 − 𝑋) · 𝑋))
65		absresq 15276	. . . . . . 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 7429	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)) = (((1 / 2)↑2) − (((1 / 2) − 𝑋)↑2)))
68	57, 64, 67	3eqtr4d 2775	. . . 4 ⊢ (𝜑 → ((abs‘(1 − 𝑋)) · (abs‘𝑋)) = (((1 / 2)↑2) − ((abs‘((1 / 2) − 𝑋))↑2)))
69	68	oveq1d 7428	. . 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 12315	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
71		2ne0 12341	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
72	71	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
73	9, 70, 72	divcan4d 12021	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
74	9	times2d 12481	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
75	74	oveq1d 7428	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
76	73, 75	eqtr3d 2767	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
77		chordthmlem4.M	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
78	76, 77	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
79	9, 9	addcld 11258	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
80	10, 9	addcld 11258	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
81	79, 80, 70, 72	divsubdird 12054	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
82	9, 10, 9	pnpcan2d 11634	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
83	82	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
84	78, 81, 83	3eqtr2d 2771	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
85	11, 70, 72	divrec2d 12019	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
86	84, 85	eqtrd 2765	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
87	86	fveq2d 6894	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = (abs‘((1 / 2) · (𝐵 − 𝐴))))
88	39, 11	absmuld 15428	. . . . . . 7 ⊢ (𝜑 → (abs‘((1 / 2) · (𝐵 − 𝐴))) = ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))))
89		0red 11242	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
90		halfgt0 12453	. . . . . . . . . . 11 ⊢ 0 < (1 / 2)
91	90	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 / 2))
92	89, 41, 91	ltled 11387	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (1 / 2))
93	41, 92	absidd 15396	. . . . . . . 8 ⊢ (𝜑 → (abs‘(1 / 2)) = (1 / 2))
94	93	oveq1d 7428	. . . . . . 7 ⊢ (𝜑 → ((abs‘(1 / 2)) · (abs‘(𝐵 − 𝐴))) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
95	87, 88, 94	3eqtrd 2769	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐵 − 𝑀)) = ((1 / 2) · (abs‘(𝐵 − 𝐴))))
96	95	oveq1d 7428	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2))
97	39, 13	sqmuld 14149	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (abs‘(𝐵 − 𝐴)))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
98	96, 97	eqtrd 2765	. . . 4 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑀))↑2) = (((1 / 2)↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
99	39, 14, 11	subdird 11696	. . . . . . . . 9 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
100	86, 30	oveq12d 7431	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
101	80	halfcld 12482	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
102	77, 101	eqeltrd 2825	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
103	9, 102, 22	nnncan1d 11630	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
104	99, 100, 103	3eqtr2rd 2772	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
105	104	fveq2d 6894	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
106	43, 11	absmuld 15428	. . . . . . 7 ⊢ (𝜑 → (abs‘(((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
107	105, 106	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (abs‘(𝑃 − 𝑀)) = ((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴))))
108	107	oveq1d 7428	. . . . 5 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2))
109	45, 13	sqmuld 14149	. . . . 5 ⊢ (𝜑 → (((abs‘((1 / 2) − 𝑋)) · (abs‘(𝐵 − 𝐴)))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
110	108, 109	eqtrd 2765	. . . 4 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑀))↑2) = (((abs‘((1 / 2) − 𝑋))↑2) · ((abs‘(𝐵 − 𝐴))↑2)))
111	98, 110	oveq12d 7431	. . 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 2776	. 2 ⊢ (𝜑 → (((abs‘(𝐵 − 𝑀))↑2) − ((abs‘(𝑃 − 𝑀))↑2)) = (((abs‘(1 − 𝑋)) · (abs‘𝑋)) · ((abs‘(𝐵 − 𝐴))↑2)))
113	37, 112	eqtr4d 2768	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 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5144 ‘cfv 6543 (class class class)co 7413 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 < clt 11273 ≤ cle 11274 − cmin 11469 / cdiv 11896 2c2 12292 [,]cicc 13354 ↑cexp 14053 abscabs 15208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-icc 13358 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210

This theorem is referenced by: chordthmlem5 26781

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