Theorem eff1i 10098

Description: The exponential function maps the set S, of complex numbers with imaginary part in a closed-below, open-above real interval of length 2 x. pi starting at A, one-to-one to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)

Hypotheses

Ref	Expression
eff1i.1	|- A e. RR
eff1i.2	|- D = (A[,)(A + (2 x. pi)))
eff1i.3	|- S = {v e. CC | (Im` v) e. D}
eff1i.4	|- F = {<.z, w>. | (z e. D /\ w = (exp` (_i x. z)))}
eff1i.5	|- C = {v e. CC | (abs` v) = 1}

Assertion

Ref	Expression
eff1i	|- (exp |` S):S-1-1->(CC \ {0})

Distinct variable groups:   v,A,w,z   v,D,w,z   w,C,z   v,F

Proof of Theorem eff1i

Step	Hyp	Ref	Expression
1		dff13 4850	. 2 |- ((exp |` S):S-1-1->(CC \ {0}) <-> ((exp |` S):S-->(CC \ {0}) /\ A.x e. S A.y e. S (((exp |` S)` x) = ((exp |` S)` y) -> x = y)))
2		eff2 8632	. . 3 |- exp:CC-->(CC \ {0})
3		fveq2 4681	. . . . . . 7 |- (v = x -> (Im` v) = (Im` x))
4	3	eleq1d 1963	. . . . . 6 |- (v = x -> ((Im` v) e. D <-> (Im` x) e. D))
5		eff1i.3	. . . . . 6 |- S = {v e. CC | (Im` v) e. D}
6	4, 5	elrab2 2416	. . . . 5 |- (x e. S <-> (x e. CC /\ (Im` x) e. D))
7	6	simplbi 349	. . . 4 |- (x e. S -> x e. CC)
8	7	ssriv 2621	. . 3 |- S C_ CC
9		fssres 4582	. . 3 |- ((exp:CC-->(CC \ {0}) /\ S C_ CC) -> (exp |` S):S-->(CC \ {0}))
10	2, 8, 9	mp2an 761	. 2 |- (exp |` S):S-->(CC \ {0})
11		fvres 4691	. . . . 5 |- (x e. S -> ((exp |` S)` x) = (exp` x))
12		fvres 4691	. . . . 5 |- (y e. S -> ((exp |` S)` y) = (exp` y))
13	11, 12	eqeqan12d 1901	. . . 4 |- ((x e. S /\ y e. S) -> (((exp |` S)` x) = ((exp |` S)` y) <-> (exp` x) = (exp` y)))
14		absef 8749	. . . . . . . . . . 11 |- (x e. CC -> (abs` (exp` x)) = (exp` (Re` x)))
15		absef 8749	. . . . . . . . . . 11 |- (y e. CC -> (abs` (exp` y)) = (exp` (Re` y)))
16	14, 15	eqeqan12d 1901	. . . . . . . . . 10 |- ((x e. CC /\ y e. CC) -> ((abs` (exp` x)) = (abs` (exp` y)) <-> (exp` (Re` x)) = (exp` (Re` y))))
17		reef11 8674	. . . . . . . . . . 11 |- (((Re` x) e. RR /\ (Re` y) e. RR) -> ((exp` (Re` x)) = (exp` (Re` y)) <-> (Re` x) = (Re` y)))
18		recl 8007	. . . . . . . . . . 11 |- (x e. CC -> (Re` x) e. RR)
19		recl 8007	. . . . . . . . . . 11 |- (y e. CC -> (Re` y) e. RR)
20	17, 18, 19	syl2an 503	. . . . . . . . . 10 |- ((x e. CC /\ y e. CC) -> ((exp` (Re` x)) = (exp` (Re` y)) <-> (Re` x) = (Re` y)))
21	16, 20	bitrd 587	. . . . . . . . 9 |- ((x e. CC /\ y e. CC) -> ((abs` (exp` x)) = (abs` (exp` y)) <-> (Re` x) = (Re` y)))
22		fveq2 4681	. . . . . . . . 9 |- ((exp` x) = (exp` y) -> (abs` (exp` x)) = (abs` (exp` y)))
23	21, 22	syl5bi 225	. . . . . . . 8 |- ((x e. CC /\ y e. CC) -> ((exp` x) = (exp` y) -> (Re` x) = (Re` y)))
24		fveq2 4681	. . . . . . . . . . 11 |- (v = y -> (Im` v) = (Im` y))
25	24	eleq1d 1963	. . . . . . . . . 10 |- (v = y -> ((Im` v) e. D <-> (Im` y) e. D))
26	25, 5	elrab2 2416	. . . . . . . . 9 |- (y e. S <-> (y e. CC /\ (Im` y) e. D))
27	26	simplbi 349	. . . . . . . 8 |- (y e. S -> y e. CC)
28	23, 7, 27	syl2an 503	. . . . . . 7 |- ((x e. S /\ y e. S) -> ((exp` x) = (exp` y) -> (Re` x) = (Re` y)))
29	28	imp 377	. . . . . 6 |- (((x e. S /\ y e. S) /\ (exp` x) = (exp` y)) -> (Re` x) = (Re` y))
30		eff1lem 10097	. . . . . . . . . . . . 13 |- (x e. CC -> (exp` x) = ((abs` (exp` x)) x. (exp` (_i x. (Im` x)))))
31		eff1lem 10097	. . . . . . . . . . . . 13 |- (y e. CC -> (exp` y) = ((abs` (exp` y)) x. (exp` (_i x. (Im` y)))))
32	30, 31	eqeqan12d 1901	. . . . . . . . . . . 12 |- ((x e. CC /\ y e. CC) -> ((exp` x) = (exp` y) <-> ((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` y)) x. (exp` (_i x. (Im` y))))))
33	32	biimpa 460	. . . . . . . . . . 11 |- (((x e. CC /\ y e. CC) /\ (exp` x) = (exp` y)) -> ((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` y)) x. (exp` (_i x. (Im` y)))))
34	22	opreq1d 4897	. . . . . . . . . . . . . 14 |- ((exp` x) = (exp` y) -> ((abs` (exp` x)) x. (exp` (_i x. (Im` y)))) = ((abs` (exp` y)) x. (exp` (_i x. (Im` y)))))
35	34	adantl 424	. . . . . . . . . . . . 13 |- (((x e. CC /\ y e. CC) /\ (exp` x) = (exp` y)) -> ((abs` (exp` x)) x. (exp` (_i x. (Im` y)))) = ((abs` (exp` y)) x. (exp` (_i x. (Im` y)))))
36	35	eqeq2d 1895	. . . . . . . . . . . 12 |- (((x e. CC /\ y e. CC) /\ (exp` x) = (exp` y)) -> (((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` x)) x. (exp` (_i x. (Im` y)))) <-> ((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` y)) x. (exp` (_i x. (Im` y))))))
37		imcl 8008	. . . . . . . . . . . . . . . . . 18 |- (x e. CC -> (Im` x) e. RR)
38	37	recnd 6468	. . . . . . . . . . . . . . . . 17 |- (x e. CC -> (Im` x) e. CC)
39		axicn 6423	. . . . . . . . . . . . . . . . . 18 |- _i e. CC
40		mulcl 6456	. . . . . . . . . . . . . . . . . 18 |- ((_i e. CC /\ (Im` x) e. CC) -> (_i x. (Im` x)) e. CC)
41	39, 40	mpan 759	. . . . . . . . . . . . . . . . 17 |- ((Im` x) e. CC -> (_i x. (Im` x)) e. CC)
42	38, 41	syl 12	. . . . . . . . . . . . . . . 16 |- (x e. CC -> (_i x. (Im` x)) e. CC)
43		efcl 8574	. . . . . . . . . . . . . . . 16 |- ((_i x. (Im` x)) e. CC -> (exp` (_i x. (Im` x))) e. CC)
44	42, 43	syl 12	. . . . . . . . . . . . . . 15 |- (x e. CC -> (exp` (_i x. (Im` x))) e. CC)
45	44	adantr 425	. . . . . . . . . . . . . 14 |- ((x e. CC /\ y e. CC) -> (exp` (_i x. (Im` x))) e. CC)
46		imcl 8008	. . . . . . . . . . . . . . . . . 18 |- (y e. CC -> (Im` y) e. RR)
47	46	recnd 6468	. . . . . . . . . . . . . . . . 17 |- (y e. CC -> (Im` y) e. CC)
48		mulcl 6456	. . . . . . . . . . . . . . . . . 18 |- ((_i e. CC /\ (Im` y) e. CC) -> (_i x. (Im` y)) e. CC)
49	39, 48	mpan 759	. . . . . . . . . . . . . . . . 17 |- ((Im` y) e. CC -> (_i x. (Im` y)) e. CC)
50	47, 49	syl 12	. . . . . . . . . . . . . . . 16 |- (y e. CC -> (_i x. (Im` y)) e. CC)
51		efcl 8574	. . . . . . . . . . . . . . . 16 |- ((_i x. (Im` y)) e. CC -> (exp` (_i x. (Im` y))) e. CC)
52	50, 51	syl 12	. . . . . . . . . . . . . . 15 |- (y e. CC -> (exp` (_i x. (Im` y))) e. CC)
53	52	adantl 424	. . . . . . . . . . . . . 14 |- ((x e. CC /\ y e. CC) -> (exp` (_i x. (Im` y))) e. CC)
54		efcl 8574	. . . . . . . . . . . . . . . . 17 |- (x e. CC -> (exp` x) e. CC)
55		abscl 8084	. . . . . . . . . . . . . . . . 17 |- ((exp` x) e. CC -> (abs` (exp` x)) e. RR)
56	54, 55	syl 12	. . . . . . . . . . . . . . . 16 |- (x e. CC -> (abs` (exp` x)) e. RR)
57	56	recnd 6468	. . . . . . . . . . . . . . 15 |- (x e. CC -> (abs` (exp` x)) e. CC)
58	57	adantr 425	. . . . . . . . . . . . . 14 |- ((x e. CC /\ y e. CC) -> (abs` (exp` x)) e. CC)
59		efne0 8631	. . . . . . . . . . . . . . . . 17 |- (x e. CC -> (exp` x) =/= 0)
60		absgt0 8145	. . . . . . . . . . . . . . . . . 18 |- ((exp` x) e. CC -> ((exp` x) =/= 0 <-> 0 < (abs` (exp` x))))
61	60	biimpa 460	. . . . . . . . . . . . . . . . 17 |- (((exp` x) e. CC /\ (exp` x) =/= 0) -> 0 < (abs` (exp` x)))
62	54, 59, 61	syl11anc 524	. . . . . . . . . . . . . . . 16 |- (x e. CC -> 0 < (abs` (exp` x)))
63		gt0ne0 6800	. . . . . . . . . . . . . . . 16 |- (((abs` (exp` x)) e. RR /\ 0 < (abs` (exp` x))) -> (abs` (exp` x)) =/= 0)
64	56, 62, 63	syl11anc 524	. . . . . . . . . . . . . . 15 |- (x e. CC -> (abs` (exp` x)) =/= 0)
65	64	adantr 425	. . . . . . . . . . . . . 14 |- ((x e. CC /\ y e. CC) -> (abs` (exp` x)) =/= 0)
66		mulcan 6880	. . . . . . . . . . . . . 14 |- (((exp` (_i x. (Im` x))) e. CC /\ (exp` (_i x. (Im` y))) e. CC /\ ((abs` (exp` x)) e. CC /\ (abs` (exp` x)) =/= 0)) -> (((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` x)) x. (exp` (_i x. (Im` y)))) <-> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y)))))
67	45, 53, 58, 65, 66	syl112anc 1104	. . . . . . . . . . . . 13 |- ((x e. CC /\ y e. CC) -> (((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` x)) x. (exp` (_i x. (Im` y)))) <-> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y)))))
68	67	adantr 425	. . . . . . . . . . . 12 |- (((x e. CC /\ y e. CC) /\ (exp` x) = (exp` y)) -> (((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` x)) x. (exp` (_i x. (Im` y)))) <-> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y)))))
69	36, 68	bitr3d 589	. . . . . . . . . . 11 |- (((x e. CC /\ y e. CC) /\ (exp` x) = (exp` y)) -> (((abs` (exp` x)) x. (exp` (_i x. (Im` x)))) = ((abs` (exp` y)) x. (exp` (_i x. (Im` y)))) <-> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y)))))
70	33, 69	mpbid 212	. . . . . . . . . 10 |- (((x e. CC /\ y e. CC) /\ (exp` x) = (exp` y)) -> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y))))
71	70	ex 402	. . . . . . . . 9 |- ((x e. CC /\ y e. CC) -> ((exp` x) = (exp` y) -> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y)))))
72	71, 7, 27	syl2an 503	. . . . . . . 8 |- ((x e. S /\ y e. S) -> ((exp` x) = (exp` y) -> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y)))))
73		opreq2 4890	. . . . . . . . . . . . 13 |- (z = (Im` x) -> (_i x. z) = (_i x. (Im` x)))
74	73	fveq2d 4685	. . . . . . . . . . . 12 |- (z = (Im` x) -> (exp` (_i x. z)) = (exp` (_i x. (Im` x))))
75		eff1i.4	. . . . . . . . . . . 12 |- F = {<.z, w>. | (z e. D /\ w = (exp` (_i x. z)))}
76		fvex 4689	. . . . . . . . . . . 12 |- (exp` (_i x. (Im` x))) e. _V
77	74, 75, 76	fvopab4 4743	. . . . . . . . . . 11 |- ((Im` x) e. D -> (F` (Im` x)) = (exp` (_i x. (Im` x))))
78		opreq2 4890	. . . . . . . . . . . . 13 |- (z = (Im` y) -> (_i x. z) = (_i x. (Im` y)))
79	78	fveq2d 4685	. . . . . . . . . . . 12 |- (z = (Im` y) -> (exp` (_i x. z)) = (exp` (_i x. (Im` y))))
80		fvex 4689	. . . . . . . . . . . 12 |- (exp` (_i x. (Im` y))) e. _V
81	79, 75, 80	fvopab4 4743	. . . . . . . . . . 11 |- ((Im` y) e. D -> (F` (Im` y)) = (exp` (_i x. (Im` y))))
82	77, 81	eqeqan12d 1901	. . . . . . . . . 10 |- (((Im` x) e. D /\ (Im` y) e. D) -> ((F` (Im` x)) = (F` (Im` y)) <-> (exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y)))))
83		eff1i.1	. . . . . . . . . . . . 13 |- A e. RR
84		eff1i.2	. . . . . . . . . . . . . 14 |- D = (A[,)(A + (2 x. pi)))
85		eff1i.5	. . . . . . . . . . . . . 14 |- C = {v e. CC | (abs` v) = 1}
86	84, 75, 85	shftefif1o 10096	. . . . . . . . . . . . 13 |- (A e. RR -> F:D-1-1-onto->C)
87	83, 86	ax-mp 7	. . . . . . . . . . . 12 |- F:D-1-1-onto->C
88		f1of1 4634	. . . . . . . . . . . 12 |- (F:D-1-1-onto->C -> F:D-1-1->C)
89	87, 88	ax-mp 7	. . . . . . . . . . 11 |- F:D-1-1->C
90		f1fveq 4852	. . . . . . . . . . 11 |- ((F:D-1-1->C /\ ((Im` x) e. D /\ (Im` y) e. D)) -> ((F` (Im` x)) = (F` (Im` y)) <-> (Im` x) = (Im` y)))
91	89, 90	mpan 759	. . . . . . . . . 10 |- (((Im` x) e. D /\ (Im` y) e. D) -> ((F` (Im` x)) = (F` (Im` y)) <-> (Im` x) = (Im` y)))
92	82, 91	bitr3d 589	. . . . . . . . 9 |- (((Im` x) e. D /\ (Im` y) e. D) -> ((exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y))) <-> (Im` x) = (Im` y)))
93	6	simprbi 353	. . . . . . . . 9 |- (x e. S -> (Im` x) e. D)
94	26	simprbi 353	. . . . . . . . 9 |- (y e. S -> (Im` y) e. D)
95	92, 93, 94	syl2an 503	. . . . . . . 8 |- ((x e. S /\ y e. S) -> ((exp` (_i x. (Im` x))) = (exp` (_i x. (Im` y))) <-> (Im` x) = (Im` y)))
96	72, 95	sylibd 219	. . . . . . 7 |- ((x e. S /\ y e. S) -> ((exp` x) = (exp` y) -> (Im` x) = (Im` y)))
97	96	imp 377	. . . . . 6 |- (((x e. S /\ y e. S) /\ (exp` x) = (exp` y)) -> (Im` x) = (Im` y))
98		replim 8011	. . . . . . . . . . 11 |- (x e. CC -> x = ((Re` x) + (_i x. (Im` x))))
99		replim 8011	. . . . . . . . . . 11 |- (y e. CC -> y = ((Re` y) + (_i x. (Im` y))))
100	98, 99	eqeqan12d 1901	. . . . . . . . . 10 |- ((x e. CC /\ y e. CC) -> (x = y <-> ((Re` x) + (_i x. (Im` x))) = ((Re` y) + (_i x. (Im` y)))))
101		cru 7988	. . . . . . . . . . 11 |- ((((Re` x) e. RR /\ (Im` x) e. RR) /\ ((Re` y) e. RR /\ (Im` y) e. RR)) -> (((Re` x) + (_i x. (Im` x))) = ((Re` y) + (_i x. (Im` y))) <-> ((Re` x) = (Re` y) /\ (Im` x) = (Im` y))))
102	18, 37	jca 310	. . . . . . . . . . 11 |- (x e. CC -> ((Re` x) e. RR /\ (Im` x) e. RR))
103	19, 46	jca 310	. . . . . . . . . . 11 |- (y e. CC -> ((Re` y) e. RR /\ (Im` y) e. RR))
104	101, 102, 103	syl2an 503	. . . . . . . . . 10 |- ((x e. CC /\ y e. CC) -> (((Re` x) + (_i x. (Im` x))) = ((Re` y) + (_i x. (Im` y))) <-> ((Re` x) = (Re` y) /\ (Im` x) = (Im` y))))
105	100, 104	bitrd 587	. . . . . . . . 9 |- ((x e. CC /\ y e. CC) -> (x = y <-> ((Re` x) = (Re` y) /\ (Im` x) = (Im` y))))
106	105	biimprd 171	. . . . . . . 8 |- ((x e. CC /\ y e. CC) -> (((Re` x) = (Re` y) /\ (Im` x) = (Im` y)) -> x = y))
107	106, 7, 27	syl2an 503	. . . . . . 7 |- ((x e. S /\ y e. S) -> (((Re` x) = (Re` y) /\ (Im` x) = (Im` y)) -> x = y))
108	107	adantr 425	. . . . . 6 |- (((x e. S /\ y e. S) /\ (exp` x) = (exp` y)) -> (((Re` x) = (Re` y) /\ (Im` x) = (Im` y)) -> x = y))
109	29, 97, 108	mp2and 767	. . . . 5 |- (((x e. S /\ y e. S) /\ (exp` x) = (exp` y)) -> x = y)
110	109	ex 402	. . . 4 |- ((x e. S /\ y e. S) -> ((exp` x) = (exp` y) -> x = y))
111	13, 110	sylbid 220	. . 3 |- ((x e. S /\ y e. S) -> (((exp |` S)` x) = ((exp |` S)` y) -> x = y))
112	111	rgen2a 2160	. 2 |- A.x e. S A.y e. S (((exp |` S)` x) = ((exp |` S)` y) -> x = y)
113	1, 10, 112	mpbir2an 800	1 |- (exp |` S):S-1-1->(CC \ {0})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  {crab 2108   \ cdif 2590   C_ wss 2593  {csn 3044   class class class wbr 3338  {copab 3395   |` cres 3988  -->wf 3994  -1-1->wf1 3995  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   < clt 6653  2c2 7145  [,)cico 7526  Recre 7997  Imcim 7998  abscabs 8000  expce 8555  picpi 8559

This theorem is referenced by:  eff1oi 10100

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-9 7161  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-ioc 7529  df-ico 7530  df-icc 7531  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-cncf 8525  df-ef 8560  df-sin 8562  df-cos 8563  df-pi 8564  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-subg 9424

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