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Theorem ftalem7 20422
Description: Lemma for fta 20423. 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 2358 . . . 4  |-  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )  =  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
2 eqid 2358 . . . 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 8941 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( z  +  X )  e.  CC )
7 cnex 8905 . . . . . . . . 9  |-  CC  e.  _V
87a1i 10 . . . . . . . 8  |-  ( ph  ->  CC  e.  _V )
94negcld 9231 . . . . . . . . 9  |-  ( ph  -> 
-u X  e.  CC )
109adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  -u X  e.  CC )
11 df-idp 19669 . . . . . . . . . 10  |-  X p  =  (  _I  |`  CC )
12 mptresid 5083 . . . . . . . . . 10  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
1311, 12eqtr4i 2381 . . . . . . . . 9  |-  X p  =  ( z  e.  CC  |->  z )
1413a1i 10 . . . . . . . 8  |-  ( ph  ->  X p  =  ( z  e.  CC  |->  z ) )
15 fconstmpt 4811 . . . . . . . . 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 6179 . . . . . . 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 9183 . . . . . . . . 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 4185 . . . . . . 7  |-  ( ph  ->  ( z  e.  CC  |->  ( z  -  -u X
) )  =  ( z  e.  CC  |->  ( z  +  X ) ) )
2217, 21eqtrd 2390 . . . . . 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 19678 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2523, 24syl 15 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
2625feqmptd 5655 . . . . . 6  |-  ( ph  ->  F  =  ( y  e.  CC  |->  ( F `
 y ) ) )
27 fveq2 5605 . . . . . 6  |-  ( y  =  ( z  +  X )  ->  ( F `  y )  =  ( F `  ( z  +  X
) ) )
286, 22, 26, 27fmptco 5771 . . . . 5  |-  ( ph  ->  ( F  o.  (
X p  o F  -  ( CC  X.  { -u X } ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
29 plyssc 19680 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
3029, 23sseldi 3254 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
31 eqid 2358 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { -u X } ) )  =  ( X p  o F  -  ( CC  X.  { -u X }
) )
3231plyremlem 19782 . . . . . . . 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 8906 . . . . . . 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 8908 . . . . . . 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 19721 . . . . 5  |-  ( ph  ->  ( F  o.  (
X p  o F  -  ( CC  X.  { -u X } ) ) )  e.  (Poly `  CC ) )
4028, 39eqeltrrd 2433 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) )  e.  (Poly `  CC ) )
4128fveq2d 5609 . . . . 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 2358 . . . . . . 7  |-  (deg `  ( X p  o F  -  ( CC  X.  { -u X }
) ) )  =  (deg `  ( X p  o F  -  ( CC  X.  { -u X } ) ) )
4442, 43, 30, 34dgrco 19754 . . . . . 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 9844 . . . . . . . 8  |-  1  e.  NN
4846, 47syl6eqel 2446 . . . . . . 7  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { -u X } ) ) )  e.  NN )
4945, 48nnmulcld 9880 . . . . . 6  |-  ( ph  ->  ( N  x.  (deg `  ( X p  o F  -  ( CC  X.  { -u X }
) ) ) )  e.  NN )
5044, 49eqeltrd 2432 . . . . 5  |-  ( ph  ->  (deg `  ( F  o.  ( X p  o F  -  ( CC  X.  { -u X }
) ) ) )  e.  NN )
5141, 50eqeltrrd 2433 . . . 4  |-  ( ph  ->  (deg `  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) )  e.  NN )
52 0cn 8918 . . . . . . 7  |-  0  e.  CC
53 oveq1 5949 . . . . . . . . 9  |-  ( z  =  0  ->  (
z  +  X )  =  ( 0  +  X ) )
5453fveq2d 5609 . . . . . . . 8  |-  ( z  =  0  ->  ( F `  ( z  +  X ) )  =  ( F `  (
0  +  X ) ) )
55 eqid 2358 . . . . . . . 8  |-  ( z  e.  CC  |->  ( F `
 ( z  +  X ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) )
56 fvex 5619 . . . . . . . 8  |-  ( F `
 ( 0  +  X ) )  e. 
_V
5754, 55, 56fvmpt 5682 . . . . . . 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 9100 . . . . . . 7  |-  ( ph  ->  ( 0  +  X
)  =  X )
6059fveq2d 5609 . . . . . 6  |-  ( ph  ->  ( F `  (
0  +  X ) )  =  ( F `
 X ) )
6158, 60syl5eq 2402 . . . . 5  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =  ( F `  X ) )
62 ftalem7.6 . . . . 5  |-  ( ph  ->  ( F `  X
)  =/=  0 )
6361, 62eqnetrd 2539 . . . 4  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =/=  0
)
641, 2, 40, 51, 63ftalem6 20421 . . 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 8906 . . . . . 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 5949 . . . . . . . . . . . 12  |-  ( z  =  y  ->  (
z  +  X )  =  ( y  +  X ) )
6968fveq2d 5609 . . . . . . . . . . 11  |-  ( z  =  y  ->  ( F `  ( z  +  X ) )  =  ( F `  (
y  +  X ) ) )
70 fvex 5619 . . . . . . . . . . 11  |-  ( F `
 ( y  +  X ) )  e. 
_V
7169, 55, 70fvmpt 5682 . . . . . . . . . 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 5609 . . . . . . . 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 5609 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  =  ( abs `  ( F `  X )
) )
7673, 75breq12d 4115 . . . . . . 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 5743 . . . . . . . . . 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 12008 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  (
y  +  X ) ) )  e.  RR )
81 ffvelrn 5743 . . . . . . . . . . 11  |-  ( ( F : CC --> CC  /\  X  e.  CC )  ->  ( F `  X
)  e.  CC )
8225, 4, 81syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  CC )
8382abscld 12008 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( F `  X )
)  e.  RR )
8483adantr 451 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  X
) )  e.  RR )
8580, 84ltnled 9053 . . . . . . 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 5605 . . . . . . . . 9  |-  ( x  =  ( y  +  X )  ->  ( F `  x )  =  ( F `  ( y  +  X
) ) )
8988fveq2d 5609 . . . . . . . 8  |-  ( x  =  ( y  +  X )  ->  ( abs `  ( F `  x ) )  =  ( abs `  ( F `  ( y  +  X ) ) ) )
9089breq2d 4114 . . . . . . 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 2960 . . . . 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 1363 . . . 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 2743 . . 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 2630 . 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 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   _Vcvv 2864   {csn 3716   class class class wbr 4102    e. cmpt 4156    _I cid 4383    X. cxp 4766   `'ccnv 4767    |` cres 4770   "cima 4771    o. ccom 4772   -->wf 5330   ` cfv 5334  (class class class)co 5942    o Fcof 6160   CCcc 8822   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827    x. cmul 8829    < clt 8954    <_ cle 8955    - cmin 9124   -ucneg 9125   NNcn 9833   abscabs 11809  Polycply 19664   X pcidp 19665  coeffccoe 19666  degcdgr 19667
This theorem is referenced by:  fta  20423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-ioc 10750  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-fl 11014  df-mod 11063  df-seq 11136  df-exp 11195  df-fac 11379  df-bc 11406  df-hash 11428  df-shft 11652  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-limsup 12035  df-clim 12052  df-rlim 12053  df-sum 12250  df-ef 12440  df-sin 12442  df-cos 12443  df-pi 12445  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-hom 13323  df-cco 13324  df-rest 13420  df-topn 13421  df-topgen 13437  df-pt 13438  df-prds 13441  df-xrs 13496  df-0g 13497  df-gsum 13498  df-qtop 13503  df-imas 13504  df-xps 13506  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-submnd 14509  df-mulg 14585  df-cntz 14886  df-cmn 15184  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lp 16968  df-perf 16969  df-cn 17057  df-cnp 17058  df-haus 17143  df-tx 17357  df-hmeo 17546  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-tms 17983  df-cncf 18479  df-0p 19123  df-limc 19314  df-dv 19315  df-ply 19668  df-idp 19669  df-coe 19670  df-dgr 19671  df-log 20015  df-cxp 20016
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