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Theorem ftalem7 22415
Description: Lemma for fta 22416. 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 2442 . . . 4  |-  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )  =  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
2 eqid 2442 . . . 4  |-  (deg `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )  =  (deg
`  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) )
3 simpr 461 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  z  e.  CC )
4 ftalem7.5 . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
54adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  X  e.  CC )
63, 5addcld 9404 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( z  +  X )  e.  CC )
7 cnex 9362 . . . . . . . . 9  |-  CC  e.  _V
87a1i 11 . . . . . . . 8  |-  ( ph  ->  CC  e.  _V )
94negcld 9705 . . . . . . . . 9  |-  ( ph  -> 
-u X  e.  CC )
109adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  -u X  e.  CC )
11 df-idp 21656 . . . . . . . . . 10  |-  Xp  =  (  _I  |`  CC )
12 mptresid 5159 . . . . . . . . . 10  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
1311, 12eqtr4i 2465 . . . . . . . . 9  |-  Xp  =  ( z  e.  CC  |->  z )
1413a1i 11 . . . . . . . 8  |-  ( ph  ->  Xp  =  ( z  e.  CC  |->  z ) )
15 fconstmpt 4881 . . . . . . . . 9  |-  ( CC 
X.  { -u X } )  =  ( z  e.  CC  |->  -u X )
1615a1i 11 . . . . . . . 8  |-  ( ph  ->  ( CC  X.  { -u X } )  =  ( z  e.  CC  |->  -u X ) )
178, 3, 10, 14, 16offval2 6335 . . . . . . 7  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { -u X } ) )  =  ( z  e.  CC  |->  ( z  -  -u X
) ) )
18 id 22 . . . . . . . . 9  |-  ( z  e.  CC  ->  z  e.  CC )
19 subneg 9657 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  X  e.  CC )  ->  ( z  -  -u X
)  =  ( z  +  X ) )
2018, 4, 19syl2anr 478 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( z  -  -u X )  =  ( z  +  X
) )
2120mpteq2dva 4377 . . . . . . 7  |-  ( ph  ->  ( z  e.  CC  |->  ( z  -  -u X
) )  =  ( z  e.  CC  |->  ( z  +  X ) ) )
2217, 21eqtrd 2474 . . . . . 6  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { -u X } ) )  =  ( z  e.  CC  |->  ( z  +  X
) ) )
23 ftalem.3 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
24 plyf 21665 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2523, 24syl 16 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
2625feqmptd 5743 . . . . . 6  |-  ( ph  ->  F  =  ( y  e.  CC  |->  ( F `
 y ) ) )
27 fveq2 5690 . . . . . 6  |-  ( y  =  ( z  +  X )  ->  ( F `  y )  =  ( F `  ( z  +  X
) ) )
286, 22, 26, 27fmptco 5875 . . . . 5  |-  ( ph  ->  ( F  o.  (
Xp  oF  -  ( CC  X.  { -u X } ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
29 plyssc 21667 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
3029, 23sseldi 3353 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
31 eqid 2442 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { -u X } ) )  =  ( Xp  oF  -  ( CC 
X.  { -u X } ) )
3231plyremlem 21769 . . . . . . . 8  |-  ( -u X  e.  CC  ->  ( ( Xp  oF  -  ( CC 
X.  { -u X } ) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC  X.  { -u X } ) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC  X.  { -u X } ) ) " { 0 } )  =  { -u X } ) )
339, 32syl 16 . . . . . . 7  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { -u X } ) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC  X.  { -u X } ) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC  X.  { -u X } ) ) " { 0 } )  =  { -u X } ) )
3433simp1d 1000 . . . . . 6  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { -u X } ) )  e.  (Poly `  CC )
)
35 addcl 9363 . . . . . . 7  |-  ( ( z  e.  CC  /\  w  e.  CC )  ->  ( z  +  w
)  e.  CC )
3635adantl 466 . . . . . 6  |-  ( (
ph  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( z  +  w
)  e.  CC )
37 mulcl 9365 . . . . . . 7  |-  ( ( z  e.  CC  /\  w  e.  CC )  ->  ( z  x.  w
)  e.  CC )
3837adantl 466 . . . . . 6  |-  ( (
ph  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( z  x.  w
)  e.  CC )
3930, 34, 36, 38plyco 21708 . . . . 5  |-  ( ph  ->  ( F  o.  (
Xp  oF  -  ( CC  X.  { -u X } ) ) )  e.  (Poly `  CC ) )
4028, 39eqeltrrd 2517 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) )  e.  (Poly `  CC ) )
4128fveq2d 5694 . . . . 5  |-  ( ph  ->  (deg `  ( F  o.  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) ) )  =  (deg `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) ) )
42 ftalem.2 . . . . . . 7  |-  N  =  (deg `  F )
43 eqid 2442 . . . . . . 7  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) )  =  (deg `  (
Xp  oF  -  ( CC  X.  { -u X } ) ) )
4442, 43, 30, 34dgrco 21741 . . . . . 6  |-  ( ph  ->  (deg `  ( F  o.  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) ) )  =  ( N  x.  (deg `  (
Xp  oF  -  ( CC  X.  { -u X } ) ) ) ) )
45 ftalem.4 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
4633simp2d 1001 . . . . . . . 8  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { -u X } ) ) )  =  1 )
47 1nn 10332 . . . . . . . 8  |-  1  e.  NN
4846, 47syl6eqel 2530 . . . . . . 7  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { -u X } ) ) )  e.  NN )
4945, 48nnmulcld 10368 . . . . . 6  |-  ( ph  ->  ( N  x.  (deg `  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) ) )  e.  NN )
5044, 49eqeltrd 2516 . . . . 5  |-  ( ph  ->  (deg `  ( F  o.  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) ) )  e.  NN )
5141, 50eqeltrrd 2517 . . . 4  |-  ( ph  ->  (deg `  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) )  e.  NN )
52 0cn 9377 . . . . . . 7  |-  0  e.  CC
53 oveq1 6097 . . . . . . . . 9  |-  ( z  =  0  ->  (
z  +  X )  =  ( 0  +  X ) )
5453fveq2d 5694 . . . . . . . 8  |-  ( z  =  0  ->  ( F `  ( z  +  X ) )  =  ( F `  (
0  +  X ) ) )
55 eqid 2442 . . . . . . . 8  |-  ( z  e.  CC  |->  ( F `
 ( z  +  X ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) )
56 fvex 5700 . . . . . . . 8  |-  ( F `
 ( 0  +  X ) )  e. 
_V
5754, 55, 56fvmpt 5773 . . . . . . 7  |-  ( 0  e.  CC  ->  (
( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  0 )  =  ( F `  ( 0  +  X
) ) )
5852, 57ax-mp 5 . . . . . 6  |-  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  0 )  =  ( F `  ( 0  +  X
) )
594addid2d 9569 . . . . . . 7  |-  ( ph  ->  ( 0  +  X
)  =  X )
6059fveq2d 5694 . . . . . 6  |-  ( ph  ->  ( F `  (
0  +  X ) )  =  ( F `
 X ) )
6158, 60syl5eq 2486 . . . . 5  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =  ( F `  X ) )
62 ftalem7.6 . . . . 5  |-  ( ph  ->  ( F `  X
)  =/=  0 )
6361, 62eqnetrd 2625 . . . 4  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =/=  0
)
641, 2, 40, 51, 63ftalem6 22414 . . 3  |-  ( ph  ->  E. y  e.  CC  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y ) )  <  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) ) )
65 id 22 . . . . . 6  |-  ( y  e.  CC  ->  y  e.  CC )
66 addcl 9363 . . . . . 6  |-  ( ( y  e.  CC  /\  X  e.  CC )  ->  ( y  +  X
)  e.  CC )
6765, 4, 66syl2anr 478 . . . . 5  |-  ( (
ph  /\  y  e.  CC )  ->  ( y  +  X )  e.  CC )
68 oveq1 6097 . . . . . . . . . . . 12  |-  ( z  =  y  ->  (
z  +  X )  =  ( y  +  X ) )
6968fveq2d 5694 . . . . . . . . . . 11  |-  ( z  =  y  ->  ( F `  ( z  +  X ) )  =  ( F `  (
y  +  X ) ) )
70 fvex 5700 . . . . . . . . . . 11  |-  ( F `
 ( y  +  X ) )  e. 
_V
7169, 55, 70fvmpt 5773 . . . . . . . . . 10  |-  ( y  e.  CC  ->  (
( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y )  =  ( F `  ( y  +  X
) ) )
7271adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  y )  =  ( F `  ( y  +  X
) ) )
7372fveq2d 5694 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) `  y ) )  =  ( abs `  ( F `  ( y  +  X ) ) ) )
7461adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) `  0 )  =  ( F `  X ) )
7574fveq2d 5694 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  =  ( abs `  ( F `  X )
) )
7673, 75breq12d 4304 . . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  CC )  ->  F : CC
--> CC )
7877, 67ffvelrnd 5843 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( F `
 ( y  +  X ) )  e.  CC )
7978abscld 12921 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  (
y  +  X ) ) )  e.  RR )
8025, 4ffvelrnd 5843 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  CC )
8180abscld 12921 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( F `  X )
)  e.  RR )
8281adantr 465 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  X
) )  e.  RR )
8379, 82ltnled 9520 . . . . . . 7  |-  ( (
ph  /\  y  e.  CC )  ->  ( ( abs `  ( F `
 ( y  +  X ) ) )  <  ( abs `  ( F `  X )
)  <->  -.  ( abs `  ( F `  X
) )  <_  ( abs `  ( F `  ( y  +  X
) ) ) ) )
8476, 83bitrd 253 . . . . . 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 ) ) ) ) )
8584biimpd 207 . . . . 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 ) ) ) ) )
86 fveq2 5690 . . . . . . . . 9  |-  ( x  =  ( y  +  X )  ->  ( F `  x )  =  ( F `  ( y  +  X
) ) )
8786fveq2d 5694 . . . . . . . 8  |-  ( x  =  ( y  +  X )  ->  ( abs `  ( F `  x ) )  =  ( abs `  ( F `  ( y  +  X ) ) ) )
8887breq2d 4303 . . . . . . 7  |-  ( x  =  ( y  +  X )  ->  (
( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) )  <->  ( abs `  ( F `  X
) )  <_  ( abs `  ( F `  ( y  +  X
) ) ) ) )
8988notbid 294 . . . . . 6  |-  ( x  =  ( y  +  X )  ->  ( -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) )  <->  -.  ( abs `  ( F `  X ) )  <_ 
( abs `  ( F `  ( y  +  X ) ) ) ) )
9089rspcev 3072 . . . . 5  |-  ( ( ( y  +  X
)  e.  CC  /\  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  (
y  +  X ) ) ) )  ->  E. x  e.  CC  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
9167, 85, 90syl6an 545 . . . 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
) ) ) )
9291rexlimdva 2840 . . 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
) ) ) )
9364, 92mpd 15 . 2  |-  ( ph  ->  E. x  e.  CC  -.  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
94 rexnal 2725 . 2  |-  ( E. x  e.  CC  -.  ( abs `  ( F `
 X ) )  <_  ( abs `  ( F `  x )
)  <->  -.  A. x  e.  CC  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
9593, 94sylib 196 1  |-  ( ph  ->  -.  A. x  e.  CC  ( abs `  ( F `  X )
)  <_  ( abs `  ( F `  x
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   A.wral 2714   E.wrex 2715   _Vcvv 2971   {csn 3876   class class class wbr 4291    e. cmpt 4349    _I cid 4630    X. cxp 4837   `'ccnv 4838    |` cres 4841   "cima 4842    o. ccom 4843   -->wf 5413   ` cfv 5417  (class class class)co 6090    oFcof 6317   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286    < clt 9417    <_ cle 9418    - cmin 9594   -ucneg 9595   NNcn 10321   abscabs 12722  Polycply 21651   Xpcidp 21652  coeffccoe 21653  degcdgr 21654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355  df-pi 13357  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cncf 20453  df-0p 21147  df-limc 21340  df-dv 21341  df-ply 21655  df-idp 21656  df-coe 21657  df-dgr 21658  df-log 22007  df-cxp 22008
This theorem is referenced by:  fta  22416
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