Theorem effoi 10099

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, onto 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
effoi	|- (exp |` S):S-onto->(CC \ {0})

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

Proof of Theorem effoi

Step	Hyp	Ref	Expression
1		dffo3 4792	. 2 |- ((exp |` S):S-onto->(CC \ {0}) <-> ((exp |` S):S-->(CC \ {0}) /\ A.y e. (CC \ {0})E.x e. S y = ((exp |` S)` x)))
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		fveq2 4681	. . . . . . 7 |- (v = ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) -> (Im` v) = (Im` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))))
12	11	eleq1d 1963	. . . . . 6 |- (v = ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) -> ((Im` v) e. D <-> (Im` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) e. D))
13	12, 5	elrab2 2416	. . . . 5 |- (((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) e. S <-> (((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) e. CC /\ (Im` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) e. D))
14		elrp 7233	. . . . . . . . 9 |- ((abs` y) e. RR+ <-> ((abs` y) e. RR /\ 0 < (abs` y)))
15		eldifi 2730	. . . . . . . . . 10 |- (y e. (CC \ {0}) -> y e. CC)
16		abscl 8084	. . . . . . . . . 10 |- (y e. CC -> (abs` y) e. RR)
17	15, 16	syl 12	. . . . . . . . 9 |- (y e. (CC \ {0}) -> (abs` y) e. RR)
18		eldifsn 3123	. . . . . . . . . 10 |- (y e. (CC \ {0}) <-> (y e. CC /\ y =/= 0))
19		absgt0 8145	. . . . . . . . . . 11 |- (y e. CC -> (y =/= 0 <-> 0 < (abs` y)))
20	19	biimpa 460	. . . . . . . . . 10 |- ((y e. CC /\ y =/= 0) -> 0 < (abs` y))
21	18, 20	sylbi 216	. . . . . . . . 9 |- (y e. (CC \ {0}) -> 0 < (abs` y))
22	14, 17, 21	sylanbrc 527	. . . . . . . 8 |- (y e. (CC \ {0}) -> (abs` y) e. RR+)
23		reeff1o2 8692	. . . . . . . . . . 11 |- (exp |` RR):RR-1-1-onto->RR+
24		f1ocnv 4651	. . . . . . . . . . 11 |- ((exp |` RR):RR-1-1-onto->RR+ -> `'(exp |` RR):RR+-1-1-onto->RR)
25	23, 24	ax-mp 7	. . . . . . . . . 10 |- `'(exp |` RR):RR+-1-1-onto->RR
26		f1of 4635	. . . . . . . . . 10 |- (`'(exp |` RR):RR+-1-1-onto->RR -> `'(exp |` RR):RR+-->RR)
27	25, 26	ax-mp 7	. . . . . . . . 9 |- `'(exp |` RR):RR+-->RR
28	27	ffvelrni 4788	. . . . . . . 8 |- ((abs` y) e. RR+ -> (`'(exp |` RR)` (abs` y)) e. RR)
29	22, 28	syl 12	. . . . . . 7 |- (y e. (CC \ {0}) -> (`'(exp |` RR)` (abs` y)) e. RR)
30	29	recnd 6468	. . . . . 6 |- (y e. (CC \ {0}) -> (`'(exp |` RR)` (abs` y)) e. CC)
31		fveq2 4681	. . . . . . . . . . . 12 |- (v = (y / (abs` y)) -> (abs` v) = (abs` (y / (abs` y))))
32	31	eqeq1d 1892	. . . . . . . . . . 11 |- (v = (y / (abs` y)) -> ((abs` v) = 1 <-> (abs` (y / (abs` y))) = 1))
33		eff1i.5	. . . . . . . . . . 11 |- C = {v e. CC | (abs` v) = 1}
34	32, 33	elrab2 2416	. . . . . . . . . 10 |- ((y / (abs` y)) e. C <-> ((y / (abs` y)) e. CC /\ (abs` (y / (abs` y))) = 1))
35	17	recnd 6468	. . . . . . . . . . 11 |- (y e. (CC \ {0}) -> (abs` y) e. CC)
36		gt0ne0 6800	. . . . . . . . . . . 12 |- (((abs` y) e. RR /\ 0 < (abs` y)) -> (abs` y) =/= 0)
37	17, 21, 36	syl11anc 524	. . . . . . . . . . 11 |- (y e. (CC \ {0}) -> (abs` y) =/= 0)
38		divcl 6901	. . . . . . . . . . 11 |- ((y e. CC /\ (abs` y) e. CC /\ (abs` y) =/= 0) -> (y / (abs` y)) e. CC)
39	15, 35, 37, 38	syl111anc 1100	. . . . . . . . . 10 |- (y e. (CC \ {0}) -> (y / (abs` y)) e. CC)
40		absdiv 8111	. . . . . . . . . . . 12 |- ((y e. CC /\ (abs` y) e. CC /\ (abs` y) =/= 0) -> (abs` (y / (abs` y))) = ((abs` y) / (abs` (abs` y))))
41	15, 35, 37, 40	syl111anc 1100	. . . . . . . . . . 11 |- (y e. (CC \ {0}) -> (abs` (y / (abs` y))) = ((abs` y) / (abs` (abs` y))))
42		absidm 8144	. . . . . . . . . . . . 13 |- (y e. CC -> (abs` (abs` y)) = (abs` y))
43	42	opreq2d 4898	. . . . . . . . . . . 12 |- (y e. CC -> ((abs` y) / (abs` (abs` y))) = ((abs` y) / (abs` y)))
44	15, 43	syl 12	. . . . . . . . . . 11 |- (y e. (CC \ {0}) -> ((abs` y) / (abs` (abs` y))) = ((abs` y) / (abs` y)))
45		divid 6942	. . . . . . . . . . . 12 |- (((abs` y) e. CC /\ (abs` y) =/= 0) -> ((abs` y) / (abs` y)) = 1)
46	35, 37, 45	syl11anc 524	. . . . . . . . . . 11 |- (y e. (CC \ {0}) -> ((abs` y) / (abs` y)) = 1)
47	41, 44, 46	3eqtrd 1929	. . . . . . . . . 10 |- (y e. (CC \ {0}) -> (abs` (y / (abs` y))) = 1)
48	34, 39, 47	sylanbrc 527	. . . . . . . . 9 |- (y e. (CC \ {0}) -> (y / (abs` y)) e. C)
49		eff1i.1	. . . . . . . . . . . . 13 |- A e. RR
50		eff1i.2	. . . . . . . . . . . . . 14 |- D = (A[,)(A + (2 x. pi)))
51		eff1i.4	. . . . . . . . . . . . . 14 |- F = {<.z, w>. | (z e. D /\ w = (exp` (_i x. z)))}
52	50, 51, 33	shftefif1o 10096	. . . . . . . . . . . . 13 |- (A e. RR -> F:D-1-1-onto->C)
53	49, 52	ax-mp 7	. . . . . . . . . . . 12 |- F:D-1-1-onto->C
54		f1ocnv 4651	. . . . . . . . . . . 12 |- (F:D-1-1-onto->C -> `'F:C-1-1-onto->D)
55	53, 54	ax-mp 7	. . . . . . . . . . 11 |- `'F:C-1-1-onto->D
56		f1of 4635	. . . . . . . . . . 11 |- (`'F:C-1-1-onto->D -> `'F:C-->D)
57	55, 56	ax-mp 7	. . . . . . . . . 10 |- `'F:C-->D
58	57	ffvelrni 4788	. . . . . . . . 9 |- ((y / (abs` y)) e. C -> (`'F` (y / (abs` y))) e. D)
59	50	eleq2i 1961	. . . . . . . . . . 11 |- ((`'F` (y / (abs` y))) e. D <-> (`'F` (y / (abs` y))) e. (A[,)(A + (2 x. pi))))
60		2re 7163	. . . . . . . . . . . . . 14 |- 2 e. RR
61		pire 10026	. . . . . . . . . . . . . 14 |- pi e. RR
62	60, 61	remulcli 6488	. . . . . . . . . . . . 13 |- (2 x. pi) e. RR
63	49, 62	readdcli 6487	. . . . . . . . . . . 12 |- (A + (2 x. pi)) e. RR
64		elico2 7559	. . . . . . . . . . . 12 |- ((A e. RR /\ (A + (2 x. pi)) e. RR) -> ((`'F` (y / (abs` y))) e. (A[,)(A + (2 x. pi))) <-> ((`'F` (y / (abs` y))) e. RR /\ A <_ (`'F` (y / (abs` y))) /\ (`'F` (y / (abs` y))) < (A + (2 x. pi)))))
65	49, 63, 64	mp2an 761	. . . . . . . . . . 11 |- ((`'F` (y / (abs` y))) e. (A[,)(A + (2 x. pi))) <-> ((`'F` (y / (abs` y))) e. RR /\ A <_ (`'F` (y / (abs` y))) /\ (`'F` (y / (abs` y))) < (A + (2 x. pi))))
66	59, 65	bitri 190	. . . . . . . . . 10 |- ((`'F` (y / (abs` y))) e. D <-> ((`'F` (y / (abs` y))) e. RR /\ A <_ (`'F` (y / (abs` y))) /\ (`'F` (y / (abs` y))) < (A + (2 x. pi))))
67	66	simp1bi 891	. . . . . . . . 9 |- ((`'F` (y / (abs` y))) e. D -> (`'F` (y / (abs` y))) e. RR)
68	48, 58, 67	3syl 24	. . . . . . . 8 |- (y e. (CC \ {0}) -> (`'F` (y / (abs` y))) e. RR)
69	68	recnd 6468	. . . . . . 7 |- (y e. (CC \ {0}) -> (`'F` (y / (abs` y))) e. CC)
70		axicn 6423	. . . . . . . 8 |- _i e. CC
71		mulcl 6456	. . . . . . . 8 |- ((_i e. CC /\ (`'F` (y / (abs` y))) e. CC) -> (_i x. (`'F` (y / (abs` y)))) e. CC)
72	70, 71	mpan 759	. . . . . . 7 |- ((`'F` (y / (abs` y))) e. CC -> (_i x. (`'F` (y / (abs` y)))) e. CC)
73	69, 72	syl 12	. . . . . 6 |- (y e. (CC \ {0}) -> (_i x. (`'F` (y / (abs` y)))) e. CC)
74		addcl 6454	. . . . . 6 |- (((`'(exp |` RR)` (abs` y)) e. CC /\ (_i x. (`'F` (y / (abs` y)))) e. CC) -> ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) e. CC)
75	30, 73, 74	syl11anc 524	. . . . 5 |- (y e. (CC \ {0}) -> ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) e. CC)
76		crim 8020	. . . . . . 7 |- (((`'(exp |` RR)` (abs` y)) e. RR /\ (`'F` (y / (abs` y))) e. RR) -> (Im` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) = (`'F` (y / (abs` y))))
77	29, 68, 76	syl11anc 524	. . . . . 6 |- (y e. (CC \ {0}) -> (Im` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) = (`'F` (y / (abs` y))))
78	48, 58	syl 12	. . . . . 6 |- (y e. (CC \ {0}) -> (`'F` (y / (abs` y))) e. D)
79	77, 78	eqeltrd 1971	. . . . 5 |- (y e. (CC \ {0}) -> (Im` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) e. D)
80	13, 75, 79	sylanbrc 527	. . . 4 |- (y e. (CC \ {0}) -> ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) e. S)
81		fvres 4691	. . . . . 6 |- (((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) e. S -> ((exp |` S)` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) = (exp` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))))
82	80, 81	syl 12	. . . . 5 |- (y e. (CC \ {0}) -> ((exp |` S)` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) = (exp` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))))
83		efadd 8629	. . . . . . 7 |- (((`'(exp |` RR)` (abs` y)) e. CC /\ (_i x. (`'F` (y / (abs` y)))) e. CC) -> (exp` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) = ((exp` (`'(exp |` RR)` (abs` y))) x. (exp` (_i x. (`'F` (y / (abs` y)))))))
84	30, 73, 83	syl11anc 524	. . . . . 6 |- (y e. (CC \ {0}) -> (exp` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) = ((exp` (`'(exp |` RR)` (abs` y))) x. (exp` (_i x. (`'F` (y / (abs` y)))))))
85		fvres 4691	. . . . . . . . 9 |- ((`'(exp |` RR)` (abs` y)) e. RR -> ((exp |` RR)` (`'(exp |` RR)` (abs` y))) = (exp` (`'(exp |` RR)` (abs` y))))
86	29, 85	syl 12	. . . . . . . 8 |- (y e. (CC \ {0}) -> ((exp |` RR)` (`'(exp |` RR)` (abs` y))) = (exp` (`'(exp |` RR)` (abs` y))))
87		f1ocnvfv2 4855	. . . . . . . . . 10 |- (((exp |` RR):RR-1-1-onto->RR+ /\ (abs` y) e. RR+) -> ((exp |` RR)` (`'(exp |` RR)` (abs` y))) = (abs` y))
88	23, 87	mpan 759	. . . . . . . . 9 |- ((abs` y) e. RR+ -> ((exp |` RR)` (`'(exp |` RR)` (abs` y))) = (abs` y))
89	22, 88	syl 12	. . . . . . . 8 |- (y e. (CC \ {0}) -> ((exp |` RR)` (`'(exp |` RR)` (abs` y))) = (abs` y))
90	86, 89	eqtr3d 1927	. . . . . . 7 |- (y e. (CC \ {0}) -> (exp` (`'(exp |` RR)` (abs` y))) = (abs` y))
91		opreq2 4890	. . . . . . . . . . 11 |- (a = (`'F` (y / (abs` y))) -> (_i x. a) = (_i x. (`'F` (y / (abs` y)))))
92	91	fveq2d 4685	. . . . . . . . . 10 |- (a = (`'F` (y / (abs` y))) -> (exp` (_i x. a)) = (exp` (_i x. (`'F` (y / (abs` y))))))
93		eleq1 1957	. . . . . . . . . . . . . 14 |- (z = a -> (z e. D <-> a e. D))
94	93	adantr 425	. . . . . . . . . . . . 13 |- ((z = a /\ w = b) -> (z e. D <-> a e. D))
95		id 73	. . . . . . . . . . . . . 14 |- (w = b -> w = b)
96		opreq2 4890	. . . . . . . . . . . . . . 15 |- (z = a -> (_i x. z) = (_i x. a))
97	96	fveq2d 4685	. . . . . . . . . . . . . 14 |- (z = a -> (exp` (_i x. z)) = (exp` (_i x. a)))
98	95, 97	eqeqan12rd 1903	. . . . . . . . . . . . 13 |- ((z = a /\ w = b) -> (w = (exp` (_i x. z)) <-> b = (exp` (_i x. a))))
99	94, 98	anbi12d 690	. . . . . . . . . . . 12 |- ((z = a /\ w = b) -> ((z e. D /\ w = (exp` (_i x. z))) <-> (a e. D /\ b = (exp` (_i x. a)))))
100	99	cbvopabv 3404	. . . . . . . . . . 11 |- {<.z, w>. | (z e. D /\ w = (exp` (_i x. z)))} = {<.a, b>. | (a e. D /\ b = (exp` (_i x. a)))}
101	51, 100	eqtri 1908	. . . . . . . . . 10 |- F = {<.a, b>. | (a e. D /\ b = (exp` (_i x. a)))}
102		fvex 4689	. . . . . . . . . 10 |- (exp` (_i x. (`'F` (y / (abs` y))))) e. _V
103	92, 101, 102	fvopab4 4743	. . . . . . . . 9 |- ((`'F` (y / (abs` y))) e. D -> (F` (`'F` (y / (abs` y)))) = (exp` (_i x. (`'F` (y / (abs` y))))))
104	48, 58, 103	3syl 24	. . . . . . . 8 |- (y e. (CC \ {0}) -> (F` (`'F` (y / (abs` y)))) = (exp` (_i x. (`'F` (y / (abs` y))))))
105		f1ocnvfv2 4855	. . . . . . . . . 10 |- ((F:D-1-1-onto->C /\ (y / (abs` y)) e. C) -> (F` (`'F` (y / (abs` y)))) = (y / (abs` y)))
106	53, 105	mpan 759	. . . . . . . . 9 |- ((y / (abs` y)) e. C -> (F` (`'F` (y / (abs` y)))) = (y / (abs` y)))
107	48, 106	syl 12	. . . . . . . 8 |- (y e. (CC \ {0}) -> (F` (`'F` (y / (abs` y)))) = (y / (abs` y)))
108	104, 107	eqtr3d 1927	. . . . . . 7 |- (y e. (CC \ {0}) -> (exp` (_i x. (`'F` (y / (abs` y))))) = (y / (abs` y)))
109	90, 108	opreq12d 4900	. . . . . 6 |- (y e. (CC \ {0}) -> ((exp` (`'(exp |` RR)` (abs` y))) x. (exp` (_i x. (`'F` (y / (abs` y)))))) = ((abs` y) x. (y / (abs` y))))
110		divcan2 6910	. . . . . . 7 |- ((y e. CC /\ (abs` y) e. CC /\ (abs` y) =/= 0) -> ((abs` y) x. (y / (abs` y))) = y)
111	15, 35, 37, 110	syl111anc 1100	. . . . . 6 |- (y e. (CC \ {0}) -> ((abs` y) x. (y / (abs` y))) = y)
112	84, 109, 111	3eqtrd 1929	. . . . 5 |- (y e. (CC \ {0}) -> (exp` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))) = y)
113	82, 112	eqtr2d 1926	. . . 4 |- (y e. (CC \ {0}) -> y = ((exp |` S)` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))))
114		fveq2 4681	. . . . . 6 |- (x = ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) -> ((exp |` S)` x) = ((exp |` S)` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y)))))))
115	114	eqeq2d 1895	. . . . 5 |- (x = ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) -> (y = ((exp |` S)` x) <-> y = ((exp |` S)` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))))))
116	115	rcla4ev 2381	. . . 4 |- ((((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))) e. S /\ y = ((exp |` S)` ((`'(exp |` RR)` (abs` y)) + (_i x. (`'F` (y / (abs` y))))))) -> E.x e. S y = ((exp |` S)` x))
117	80, 113, 116	syl11anc 524	. . 3 |- (y e. (CC \ {0}) -> E.x e. S y = ((exp |` S)` x))
118	117	rgen 2159	. 2 |- A.y e. (CC \ {0})E.x e. S y = ((exp |` S)` x)
119	1, 10, 118	mpbir2an 800	1 |- (exp |` S):S-onto->(CC \ {0})

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   \ cdif 2590   C_ wss 2593  {csn 3044   class class class wbr 3338  {copab 3395  `'ccnv 3985   |` cres 3988  -->wf 3994  -onto->wfo 3996  -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   / cdiv 6447   <_ cle 6448  RR+crp 6453   < clt 6653  2c2 7145  [,)cico 7526  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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