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Theorem ftalem7 23108
Description: Lemma for fta 23109. 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 2467 . . . 4  |-  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )  =  (coeff `  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
2 eqid 2467 . . . 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 9615 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( z  +  X )  e.  CC )
7 cnex 9573 . . . . . . . . 9  |-  CC  e.  _V
87a1i 11 . . . . . . . 8  |-  ( ph  ->  CC  e.  _V )
94negcld 9917 . . . . . . . . 9  |-  ( ph  -> 
-u X  e.  CC )
109adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  -u X  e.  CC )
11 df-idp 22349 . . . . . . . . . 10  |-  Xp  =  (  _I  |`  CC )
12 mptresid 5328 . . . . . . . . . 10  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
1311, 12eqtr4i 2499 . . . . . . . . 9  |-  Xp  =  ( z  e.  CC  |->  z )
1413a1i 11 . . . . . . . 8  |-  ( ph  ->  Xp  =  ( z  e.  CC  |->  z ) )
15 fconstmpt 5043 . . . . . . . . 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 6540 . . . . . . 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 9868 . . . . . . . . 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 4533 . . . . . . 7  |-  ( ph  ->  ( z  e.  CC  |->  ( z  -  -u X
) )  =  ( z  e.  CC  |->  ( z  +  X ) ) )
2217, 21eqtrd 2508 . . . . . 6  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { -u X } ) )  =  ( z  e.  CC  |->  ( z  +  X
) ) )
23 ftalem.3 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
24 plyf 22358 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2523, 24syl 16 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
2625feqmptd 5920 . . . . . 6  |-  ( ph  ->  F  =  ( y  e.  CC  |->  ( F `
 y ) ) )
27 fveq2 5866 . . . . . 6  |-  ( y  =  ( z  +  X )  ->  ( F `  y )  =  ( F `  ( z  +  X
) ) )
286, 22, 26, 27fmptco 6054 . . . . 5  |-  ( ph  ->  ( F  o.  (
Xp  oF  -  ( CC  X.  { -u X } ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) ) )
29 plyssc 22360 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
3029, 23sseldi 3502 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
31 eqid 2467 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { -u X } ) )  =  ( Xp  oF  -  ( CC 
X.  { -u X } ) )
3231plyremlem 22462 . . . . . . . 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 1008 . . . . . 6  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { -u X } ) )  e.  (Poly `  CC )
)
35 addcl 9574 . . . . . . 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 9576 . . . . . . 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 22401 . . . . 5  |-  ( ph  ->  ( F  o.  (
Xp  oF  -  ( CC  X.  { -u X } ) ) )  e.  (Poly `  CC ) )
4028, 39eqeltrrd 2556 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  ( F `  ( z  +  X ) ) )  e.  (Poly `  CC ) )
4128fveq2d 5870 . . . . 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 2467 . . . . . . 7  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) )  =  (deg `  (
Xp  oF  -  ( CC  X.  { -u X } ) ) )
4442, 43, 30, 34dgrco 22434 . . . . . 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 1009 . . . . . . . 8  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { -u X } ) ) )  =  1 )
47 1nn 10547 . . . . . . . 8  |-  1  e.  NN
4846, 47syl6eqel 2563 . . . . . . 7  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { -u X } ) ) )  e.  NN )
4945, 48nnmulcld 10583 . . . . . 6  |-  ( ph  ->  ( N  x.  (deg `  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) ) )  e.  NN )
5044, 49eqeltrd 2555 . . . . 5  |-  ( ph  ->  (deg `  ( F  o.  ( Xp  oF  -  ( CC 
X.  { -u X } ) ) ) )  e.  NN )
5141, 50eqeltrrd 2556 . . . 4  |-  ( ph  ->  (deg `  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) )  e.  NN )
52 0cn 9588 . . . . . . 7  |-  0  e.  CC
53 oveq1 6291 . . . . . . . . 9  |-  ( z  =  0  ->  (
z  +  X )  =  ( 0  +  X ) )
5453fveq2d 5870 . . . . . . . 8  |-  ( z  =  0  ->  ( F `  ( z  +  X ) )  =  ( F `  (
0  +  X ) ) )
55 eqid 2467 . . . . . . . 8  |-  ( z  e.  CC  |->  ( F `
 ( z  +  X ) ) )  =  ( z  e.  CC  |->  ( F `  ( z  +  X
) ) )
56 fvex 5876 . . . . . . . 8  |-  ( F `
 ( 0  +  X ) )  e. 
_V
5754, 55, 56fvmpt 5950 . . . . . . 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 9780 . . . . . . 7  |-  ( ph  ->  ( 0  +  X
)  =  X )
6059fveq2d 5870 . . . . . 6  |-  ( ph  ->  ( F `  (
0  +  X ) )  =  ( F `
 X ) )
6158, 60syl5eq 2520 . . . . 5  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =  ( F `  X ) )
62 ftalem7.6 . . . . 5  |-  ( ph  ->  ( F `  X
)  =/=  0 )
6361, 62eqnetrd 2760 . . . 4  |-  ( ph  ->  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 )  =/=  0
)
641, 2, 40, 51, 63ftalem6 23107 . . 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 9574 . . . . . 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 6291 . . . . . . . . . . . 12  |-  ( z  =  y  ->  (
z  +  X )  =  ( y  +  X ) )
6968fveq2d 5870 . . . . . . . . . . 11  |-  ( z  =  y  ->  ( F `  ( z  +  X ) )  =  ( F `  (
y  +  X ) ) )
70 fvex 5876 . . . . . . . . . . 11  |-  ( F `
 ( y  +  X ) )  e. 
_V
7169, 55, 70fvmpt 5950 . . . . . . . . . 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 5870 . . . . . . . 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 5870 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( ( z  e.  CC  |->  ( F `  ( z  +  X
) ) ) ` 
0 ) )  =  ( abs `  ( F `  X )
) )
7673, 75breq12d 4460 . . . . . . 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 6022 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( F `
 ( y  +  X ) )  e.  CC )
7978abscld 13230 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  (
y  +  X ) ) )  e.  RR )
8025, 4ffvelrnd 6022 . . . . . . . . . 10  |-  ( ph  ->  ( F `  X
)  e.  CC )
8180abscld 13230 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( F `  X )
)  e.  RR )
8281adantr 465 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( abs `  ( F `  X
) )  e.  RR )
8379, 82ltnled 9731 . . . . . . 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 5866 . . . . . . . . 9  |-  ( x  =  ( y  +  X )  ->  ( F `  x )  =  ( F `  ( y  +  X
) ) )
8786fveq2d 5870 . . . . . . . 8  |-  ( x  =  ( y  +  X )  ->  ( abs `  ( F `  x ) )  =  ( abs `  ( F `  ( y  +  X ) ) ) )
8887breq2d 4459 . . . . . . 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 3214 . . . . 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 2955 . . 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 2912 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   {csn 4027   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    o. ccom 5003   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   -ucneg 9806   NNcn 10536   abscabs 13030  Polycply 22344   Xpcidp 22345  coeffccoe 22346  degcdgr 22347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-0p 21840  df-limc 22033  df-dv 22034  df-ply 22348  df-idp 22349  df-coe 22350  df-dgr 22351  df-log 22700  df-cxp 22701
This theorem is referenced by:  fta  23109
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