MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cxpaddle Unicode version

Theorem cxpaddle 20504
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
cxpaddle.1  |-  ( ph  ->  A  e.  RR )
cxpaddle.2  |-  ( ph  ->  0  <_  A )
cxpaddle.3  |-  ( ph  ->  B  e.  RR )
cxpaddle.4  |-  ( ph  ->  0  <_  B )
cxpaddle.5  |-  ( ph  ->  C  e.  RR+ )
cxpaddle.6  |-  ( ph  ->  C  <_  1 )
Assertion
Ref Expression
cxpaddle  |-  ( ph  ->  ( ( A  +  B )  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) )

Proof of Theorem cxpaddle
StepHypRef Expression
1 cxpaddle.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2 cxpaddle.3 . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
31, 2readdcld 9049 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  RR )
4 cxpaddle.2 . . . . . . . 8  |-  ( ph  ->  0  <_  A )
5 cxpaddle.4 . . . . . . . 8  |-  ( ph  ->  0  <_  B )
61, 2, 4, 5addge0d 9535 . . . . . . 7  |-  ( ph  ->  0  <_  ( A  +  B ) )
7 cxpaddle.5 . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
87rpred 10581 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
93, 6, 8recxpcld 20482 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
109adantr 452 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
1110recnd 9048 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  CC )
1211mulid2d 9040 . . 3  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( 1  x.  ( ( A  +  B )  ^ c  C ) )  =  ( ( A  +  B )  ^ c  C ) )
131adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  e.  RR )
143anim1i 552 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
15 elrp 10547 . . . . . . . 8  |-  ( ( A  +  B )  e.  RR+  <->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
1614, 15sylibr 204 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  RR+ )
1713, 16rerpdivcld 10608 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  e.  RR )
182adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  RR )
1918, 16rerpdivcld 10608 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  e.  RR )
204adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  A )
213adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  RR )
22 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <  ( A  +  B ) )
23 divge0 9812 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( ( A  +  B )  e.  RR  /\  0  <  ( A  +  B ) ) )  ->  0  <_  ( A  /  ( A  +  B ) ) )
2413, 20, 21, 22, 23syl22anc 1185 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  ( A  /  ( A  +  B ) ) )
258adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  RR )
2617, 24, 25recxpcld 20482 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  e.  RR )
275adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  B )
28 divge0 9812 . . . . . . . 8  |-  ( ( ( B  e.  RR  /\  0  <_  B )  /\  ( ( A  +  B )  e.  RR  /\  0  <  ( A  +  B ) ) )  ->  0  <_  ( B  /  ( A  +  B ) ) )
2918, 27, 21, 22, 28syl22anc 1185 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  ( B  /  ( A  +  B ) ) )
3019, 29, 25recxpcld 20482 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  e.  RR )
311, 2addge01d 9547 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( A  +  B ) ) )
325, 31mpbid 202 . . . . . . . . . 10  |-  ( ph  ->  A  <_  ( A  +  B ) )
3332adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  <_  ( A  +  B ) )
3421recnd 9048 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  CC )
3534mulid1d 9039 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  x.  1 )  =  ( A  +  B ) )
3633, 35breqtrrd 4180 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  <_  ( ( A  +  B
)  x.  1 ) )
37 1re 9024 . . . . . . . . . 10  |-  1  e.  RR
3837a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  e.  RR )
39 ledivmul 9816 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  (
( A  +  B
)  e.  RR  /\  0  <  ( A  +  B ) ) )  ->  ( ( A  /  ( A  +  B ) )  <_ 
1  <->  A  <_  ( ( A  +  B )  x.  1 ) ) )
4013, 38, 21, 22, 39syl112anc 1188 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  <_ 
1  <->  A  <_  ( ( A  +  B )  x.  1 ) ) )
4136, 40mpbird 224 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  <_ 
1 )
427adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  RR+ )
43 cxpaddle.6 . . . . . . . 8  |-  ( ph  ->  C  <_  1 )
4443adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  <_  1 )
4517, 24, 41, 42, 44cxpaddlelem 20503 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  <_ 
( ( A  / 
( A  +  B
) )  ^ c  C ) )
462, 1addge02d 9548 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  <_  A  <->  B  <_  ( A  +  B ) ) )
474, 46mpbid 202 . . . . . . . . . 10  |-  ( ph  ->  B  <_  ( A  +  B ) )
4847adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  <_  ( A  +  B ) )
4948, 35breqtrrd 4180 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  <_  ( ( A  +  B
)  x.  1 ) )
50 ledivmul 9816 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  1  e.  RR  /\  (
( A  +  B
)  e.  RR  /\  0  <  ( A  +  B ) ) )  ->  ( ( B  /  ( A  +  B ) )  <_ 
1  <->  B  <_  ( ( A  +  B )  x.  1 ) ) )
5118, 38, 21, 22, 50syl112anc 1188 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  <_ 
1  <->  B  <_  ( ( A  +  B )  x.  1 ) ) )
5249, 51mpbird 224 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  <_ 
1 )
5319, 29, 52, 42, 44cxpaddlelem 20503 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  <_ 
( ( B  / 
( A  +  B
) )  ^ c  C ) )
5417, 19, 26, 30, 45, 53le2addd 9577 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  <_  (
( ( A  / 
( A  +  B
) )  ^ c  C )  +  ( ( B  /  ( A  +  B )
)  ^ c  C
) ) )
5513recnd 9048 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  e.  CC )
5618recnd 9048 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  CC )
5716rpne0d 10586 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  =/=  0
)
5855, 56, 34, 57divdird 9761 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  ( ( A  / 
( A  +  B
) )  +  ( B  /  ( A  +  B ) ) ) )
5934, 57dividd 9721 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  1 )
6058, 59eqtr3d 2422 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  =  1 )
618recnd 9048 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
6261adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  CC )
6313, 20, 16, 62divcxpd 20481 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  =  ( ( A  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6418, 27, 16, 62divcxpd 20481 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  =  ( ( B  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6563, 64oveq12d 6039 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( (
( A  /  ( A  +  B )
)  ^ c  C
)  +  ( ( B  /  ( A  +  B ) )  ^ c  C ) )  =  ( ( ( A  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) )  +  ( ( B  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) ) )
661, 4, 8recxpcld 20482 . . . . . . . . 9  |-  ( ph  ->  ( A  ^ c  C )  e.  RR )
6766recnd 9048 . . . . . . . 8  |-  ( ph  ->  ( A  ^ c  C )  e.  CC )
6867adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  ^ c  C )  e.  CC )
692, 5, 8recxpcld 20482 . . . . . . . . 9  |-  ( ph  ->  ( B  ^ c  C )  e.  RR )
7069recnd 9048 . . . . . . . 8  |-  ( ph  ->  ( B  ^ c  C )  e.  CC )
7170adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  ^ c  C )  e.  CC )
7216, 25rpcxpcld 20489 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR+ )
7372rpne0d 10586 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  =/=  0 )
7468, 71, 11, 73divdird 9761 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( (
( A  ^ c  C )  +  ( B  ^ c  C
) )  /  (
( A  +  B
)  ^ c  C
) )  =  ( ( ( A  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) )  +  ( ( B  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) ) )
7565, 74eqtr4d 2423 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( (
( A  /  ( A  +  B )
)  ^ c  C
)  +  ( ( B  /  ( A  +  B ) )  ^ c  C ) )  =  ( ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  /  (
( A  +  B
)  ^ c  C
) ) )
7654, 60, 753brtr3d 4183 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  <_  ( ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  /  (
( A  +  B
)  ^ c  C
) ) )
7766, 69readdcld 9049 . . . . . 6  |-  ( ph  ->  ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  e.  RR )
7877adantr 452 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  ^ c  C )  +  ( B  ^ c  C ) )  e.  RR )
7938, 78, 72lemuldivd 10626 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( (
1  x.  ( ( A  +  B )  ^ c  C ) )  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) )  <->  1  <_  ( ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  /  (
( A  +  B
)  ^ c  C
) ) ) )
8076, 79mpbird 224 . . 3  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( 1  x.  ( ( A  +  B )  ^ c  C ) )  <_ 
( ( A  ^ c  C )  +  ( B  ^ c  C
) ) )
8112, 80eqbrtrrd 4176 . 2  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C
) ) )
827rpne0d 10586 . . . . . 6  |-  ( ph  ->  C  =/=  0 )
8361, 820cxpd 20469 . . . . 5  |-  ( ph  ->  ( 0  ^ c  C )  =  0 )
841, 4, 8cxpge0d 20483 . . . . . 6  |-  ( ph  ->  0  <_  ( A  ^ c  C )
)
852, 5, 8cxpge0d 20483 . . . . . 6  |-  ( ph  ->  0  <_  ( B  ^ c  C )
)
8666, 69, 84, 85addge0d 9535 . . . . 5  |-  ( ph  ->  0  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) ) )
8783, 86eqbrtrd 4174 . . . 4  |-  ( ph  ->  ( 0  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) )
88 oveq1 6028 . . . . 5  |-  ( 0  =  ( A  +  B )  ->  (
0  ^ c  C
)  =  ( ( A  +  B )  ^ c  C ) )
8988breq1d 4164 . . . 4  |-  ( 0  =  ( A  +  B )  ->  (
( 0  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) )  <->  ( ( A  +  B )  ^ c  C )  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C
) ) ) )
9087, 89syl5ibcom 212 . . 3  |-  ( ph  ->  ( 0  =  ( A  +  B )  ->  ( ( A  +  B )  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) ) )
9190imp 419 . 2  |-  ( (
ph  /\  0  =  ( A  +  B
) )  ->  (
( A  +  B
)  ^ c  C
)  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) ) )
92 0re 9025 . . . 4  |-  0  e.  RR
93 leloe 9095 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  +  B
)  e.  RR )  ->  ( 0  <_ 
( A  +  B
)  <->  ( 0  < 
( A  +  B
)  \/  0  =  ( A  +  B
) ) ) )
9492, 3, 93sylancr 645 . . 3  |-  ( ph  ->  ( 0  <_  ( A  +  B )  <->  ( 0  <  ( A  +  B )  \/  0  =  ( A  +  B ) ) ) )
956, 94mpbid 202 . 2  |-  ( ph  ->  ( 0  <  ( A  +  B )  \/  0  =  ( A  +  B )
) )
9681, 91, 95mpjaodan 762 1  |-  ( ph  ->  ( ( A  +  B )  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4154  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    < clt 9054    <_ cle 9055    / cdiv 9610   RR+crp 10545    ^ c ccxp 20321
This theorem is referenced by:  abvcxp  21177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322  df-cxp 20323
  Copyright terms: Public domain W3C validator