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Theorem ftalem7 20316
Description: Lemma for fta 20317. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
ftalem.1  |-  A  =  (coeff `  F )
ftalem.2  |-  N  =  (deg `  F )
ftalem.3  |-  ( ph  ->  F  e.  (Poly `  S ) )
ftalem.4  |-  ( ph  ->  N  e.  NN )
ftalem7.5  |-  ( ph  ->  X  e.  CC )
ftalem7.6  |-  ( ph  ->  ( F `  X
)  =/=  0 )
Assertion
Ref Expression
ftalem7  |-  ( ph  ->  -.  A. x  e.  CC  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
Distinct variable groups:    x, A    x, N    x, F    ph, x    x, X
Allowed substitution hint:    S( x)

Proof of Theorem ftalem7
Dummy variables  z  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )  =  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
2 eqid 2283 . . . 4  |-  (deg `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )  =  (deg
`  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) )
3 simpr 447 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  z  e.  CC )
4 ftalem7.5 . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
54adantr 451 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  X  e.  CC )
63, 5addcld 8854 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( z  +  X )  e.  CC )
7 cnex 8818 . . . . . . . . 9  |-  CC  e.  _V
87a1i 10 . . . . . . . 8  |-  ( ph  ->  CC  e.  _V )
94negcld 9144 . . . . . . . . 9  |-  ( ph  -> 
-u X  e.  CC )
109adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  -u X  e.  CC )
11 df-idp 19571 . . . . . . . . . 10  |-  X p  =  (  _I  |`  CC )
12 mptresid 5004 . . . . . . . . . 10  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
1311, 12eqtr4i 2306 . . . . . . . . 9  |-  X p  =  ( z  e.  CC  |->  z )
1413a1i 10 . . . . . . . 8  |-  ( ph  ->  X p  =  ( z  e.  CC  |->  z ) )
15 fconstmpt 4732 . . . . . . . . 9  |-  ( CC 
X.  { -u X } )  =  ( z  e.  CC  |->  -u X )
1615a1i 10 . . . . . . . 8  |-  ( ph  ->  ( CC  X.  { -u X } )  =  ( z  e.  CC  |->  -u X ) )
178, 3, 10, 14, 16offval2 6095 . . . . . . 7  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { -u X }
) )  =  ( z  e.  CC  |->  ( z  -  -u X
) ) )
18 id 19 . . . . . . . . 9  |-  ( z  e.  CC  ->  z  e.  CC )
19 subneg 9096 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  X  e.  CC )  ->  ( z  -  -u X
)  =  ( z  +  X ) )
2018, 4, 19syl2anr 464 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( z  -  -u X )  =  ( z  +  X
) )
2120mpteq2dva 4106 . . . . . . 7  |-  ( ph  ->  ( z  e.  CC  |->  ( z  -  -u X
) )  =  ( z  e.  CC  |->  ( z  +  X ) ) )
2217, 21eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { -u X }
) )  =  ( z  e.  CC  |->  ( z  +  X ) ) )
23 ftalem.3 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
24 plyf 19580 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2523, 24syl 15 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
2625feqmptd 5575 . . . . . 6  |-  ( ph  ->  F  =  ( y  e.  CC  |->  ( F `
 y ) ) )
27 fveq2 5525 . . . . . 6  |-  ( y  =  ( z  +  X )  ->  ( F `  y )  =  ( F `  ( z  +  X
) ) )
286, 22, 26, 27fmptco 5691 . . . . 5  |-  ( ph  ->  ( F  o.  (
X p  o F  -  ( CC  X.  { -u X } ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
29 plyssc 19582 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
3029, 23sseldi 3178 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
31 eqid 2283 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { -u X } ) )  =  ( X p  o F  -  ( CC  X.  { -u X }
) )
3231plyremlem 19684 . . . . . . . 8  |-  ( -u X  e.  CC  ->  ( ( X p  o F  -  ( CC  X.  { -u X }
) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { -u X }
) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { -u X }
) ) " {
0 } )  =  { -u X }
) )
339, 32syl 15 . . . . . . 7  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { -u X } ) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { -u X } ) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { -u X } ) ) " { 0 } )  =  { -u X } ) )
3433simp1d 967 . . . . . 6  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { -u X }
) )  e.  (Poly `  CC ) )
35 addcl 8819 . . . . . . 7  |-  ( ( z  e.  CC  /\  w  e.  CC )  ->  ( z  +  w
)  e.  CC )
3635adantl 452 . . . . . 6  |-  ( (
ph  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( z  +  w
)  e.  CC )
37 mulcl 8821 . . . . . . 7  |-  ( ( z  e.  CC  /\  w  e.  CC )  ->  ( z  x.  w
)  e.  CC )
3837adantl 452 . . . . . 6  |-  ( (
ph  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( z  x.  w
)  e.  CC )
3930, 34, 36, 38plyco 19623 . . . . 5  |-  ( ph  ->  ( F  o.  (
X p  o F  -  ( CC  X.  { -u X } ) ) )  e.  (Poly `  CC ) )
4028, 39eqeltrrd 2358 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) )  e.  (Poly `  CC ) )
4128fveq2d 5529 . . . . 5  |-  ( ph  ->  (deg `  ( F  o.  ( X p  o F  -  ( CC  X.  { -u X }
) ) ) )  =  (deg `  (
z  e.  CC  |->  ( F `  ( z  +  X ) ) ) ) )
42 ftalem.2 . . . . . . 7  |-  N  =  (deg `  F )
43 eqid 2283 . . . . . . 7  |-  (deg `  ( X p  o F  -  ( CC  X.  { -u X }
) ) )  =  (deg `  ( X p  o F  -  ( CC  X.  { -u X } ) ) )
4442, 43, 30, 34dgrco 19656 . . . . . 6  |-  ( ph  ->  (deg `  ( F  o.  ( X p  o F  -  ( CC  X.  { -u X }
) ) ) )  =  ( N  x.  (deg `  ( X p  o F  -  ( CC  X.  { -u X } ) ) ) ) )
45 ftalem.4 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
4633simp2d 968 . . . . . . . 8  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { -u X } ) ) )  =  1 )
47 1nn 9757 . . . . . . . 8  |-  1  e.  NN
4846, 47syl6eqel 2371 . . . . . . 7  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { -u X } ) ) )  e.  NN )
4945, 48nnmulcld 9793 . . . . . 6  |-  ( ph  ->  ( N  x.  (deg `  ( X p  o F  -  ( CC  X.  { -u X }
) ) ) )  e.  NN )
5044, 49eqeltrd 2357 . . . . 5  |-  ( ph  ->  (deg `  ( F  o.  ( X p  o F  -  ( CC  X.  { -u X }
) ) ) )  e.  NN )
5141, 50eqeltrrd 2358 . . . 4  |-  ( ph  ->  (deg `  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) )  e.  NN )
52 0cn 8831 . . . . . . 7  |-  0  e.  CC
53 oveq1 5865 . . . . . . . . 9  |-  ( z  =  0  ->  (
z  +  X )  =  ( 0  +  X ) )
5453fveq2d 5529 . . . . . . . 8  |-  ( z  =  0  ->  ( F `  ( z  +  X ) )  =  ( F `  (
0  +  X ) ) )
55 eqid 2283 . . . . . . . 8  |-  ( z  e.  CC  |->  ( F `
 ( z  +  X ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) )
56 fvex 5539 . . . . . . . 8  |-  ( F `
 ( 0  +  X ) )  e. 
_V
5754, 55, 56fvmpt 5602 . . . . . . 7  |-  ( 0  e.  CC  ->  (
( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  0 )  =  ( F `  ( 0  +  X
) ) )
5852, 57ax-mp 8 . . . . . 6  |-  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  0 )  =  ( F `  ( 0  +  X
) )
594addid2d 9013 . . . . . . 7  |-  ( ph  ->  ( 0  +  X
)  =  X )
6059fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( F `  (
0  +  X ) )  =  ( F `
 X ) )
6158, 60syl5eq 2327 . . . . 5  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =  ( F `  X ) )
62 ftalem7.6 . . . . 5  |-  ( ph  ->  ( F `  X
)  =/=  0 )
6361, 62eqnetrd 2464 . . . 4  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =/=  0
)
641, 2, 40, 51, 63ftalem6 20315 . . 3  |-  ( ph  ->  E. y  e.  CC  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y ) )  <  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) ) )
65 id 19 . . . . . 6  |-  ( y  e.  CC  ->  y  e.  CC )
66 addcl 8819 . . . . . 6  |-  ( ( y  e.  CC  /\  X  e.  CC )  ->  ( y  +  X
)  e.  CC )
6765, 4, 66syl2anr 464 . . . . 5  |-  ( (
ph  /\  y  e.  CC )  ->  ( y  +  X )  e.  CC )
68 oveq1 5865 . . . . . . . . . . . 12  |-  ( z  =  y  ->  (
z  +  X )  =  ( y  +  X ) )
6968fveq2d 5529 . . . . . . . . . . 11  |-  ( z  =  y  ->  ( F `  ( z  +  X ) )  =  ( F `  (
y  +  X ) ) )
70 fvex 5539 . . . . . . . . . . 11  |-  ( F `
 ( y  +  X ) )  e. 
_V
7169, 55, 70fvmpt 5602 . . . . . . . . . 10  |-  ( y  e.  CC  ->  (
( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y )  =  ( F `  ( y  +  X
) ) )
7271adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y )  =  ( F `  ( y  +  X
) ) )
7372fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) `  y ) )  =  ( abs `  ( F `  ( y  +  X ) ) ) )
7461adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  0 )  =  ( F `  X ) )
7574fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  =  ( abs `  ( F `  X )
) )
7673, 75breq12d 4036 . . . . . . 7  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y ) )  <  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  <->  ( abs `  ( F `  (
y  +  X ) ) )  <  ( abs `  ( F `  X ) ) ) )
7725adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  CC )  ->  F : CC
--> CC )
78 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( F : CC --> CC  /\  ( y  +  X
)  e.  CC )  ->  ( F `  ( y  +  X
) )  e.  CC )
7977, 67, 78syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( F `
 ( y  +  X ) )  e.  CC )
8079abscld 11918 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  (
y  +  X ) ) )  e.  RR )
81 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : CC --> CC  /\  X  e.  CC )  ->  ( F `  X
)  e.  CC )
8225, 4, 81syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  CC )
8382abscld 11918 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( F `  X )
)  e.  RR )
8483adantr 451 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  X
) )  e.  RR )
8580, 84ltnled 8966 . . . . . . 7  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( abs `  ( F `
 ( y  +  X ) ) )  <  ( abs `  ( F `  X )
)  <->  -.  ( abs `  ( F `  X
) )  <_  ( abs `  ( F `  ( y  +  X
) ) ) ) )
8676, 85bitrd 244 . . . . . 6  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y ) )  <  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  <->  -.  ( abs `  ( F `  X ) )  <_ 
( abs `  ( F `  ( y  +  X ) ) ) ) )
8786biimpd 198 . . . . 5  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y ) )  <  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  ->  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  (
y  +  X ) ) ) ) )
88 fveq2 5525 . . . . . . . . 9  |-  ( x  =  ( y  +  X )  ->  ( F `  x )  =  ( F `  ( y  +  X
) ) )
8988fveq2d 5529 . . . . . . . 8  |-  ( x  =  ( y  +  X )  ->  ( abs `  ( F `  x ) )  =  ( abs `  ( F `  ( y  +  X ) ) ) )
9089breq2d 4035 . . . . . . 7  |-  ( x  =  ( y  +  X )  ->  (
( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) )  <->  ( abs `  ( F `  X
) )  <_  ( abs `  ( F `  ( y  +  X
) ) ) ) )
9190notbid 285 . . . . . 6  |-  ( x  =  ( y  +  X )  ->  ( -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) )  <->  -.  ( abs `  ( F `  X ) )  <_ 
( abs `  ( F `  ( y  +  X ) ) ) ) )
9291rspcev 2884 . . . . 5  |-  ( ( ( y  +  X
)  e.  CC  /\  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  (
y  +  X ) ) ) )  ->  E. x  e.  CC  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
9367, 87, 92ee12an 1353 . . . 4  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y ) )  <  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  ->  E. x  e.  CC  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) ) )
9493rexlimdva 2667 . . 3  |-  ( ph  ->  ( E. y  e.  CC  ( abs `  (
( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y ) )  <  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  ->  E. x  e.  CC  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) ) )
9564, 94mpd 14 . 2  |-  ( ph  ->  E. x  e.  CC  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
96 rexnal 2554 . 2  |-  ( E. x  e.  CC  -.  ( abs `  ( F `
 X ) )  <_  ( abs `  ( F `  x )
)  <->  -.  A. x  e.  CC  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
9795, 96sylib 188 1  |-  ( ph  ->  -.  A. x  e.  CC  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   {csn 3640   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038   NNcn 9746   abscabs 11719  Polycply 19566   X pcidp 19567  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  fta  20317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-0p 19025  df-limc 19216  df-dv 19217  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-log 19914  df-cxp 19915
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