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Theorem cxpaddle 20086
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 8857 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  RR )
4 cxpaddle.2 . . . . . . . 8  |-  ( ph  ->  0  <_  A )
5 cxpaddle.4 . . . . . . . 8  |-  ( ph  ->  0  <_  B )
61, 2, 4, 5addge0d 9343 . . . . . . 7  |-  ( ph  ->  0  <_  ( A  +  B ) )
7 cxpaddle.5 . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
87rpred 10385 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
93, 6, 8recxpcld 20064 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
109adantr 453 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
1110recnd 8856 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  CC )
1211mulid2d 8848 . . 3  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( 1  x.  ( ( A  +  B )  ^ c  C ) )  =  ( ( A  +  B )  ^ c  C ) )
131adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  e.  RR )
143anim1i 553 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
15 elrp 10351 . . . . . . . 8  |-  ( ( A  +  B )  e.  RR+  <->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
1614, 15sylibr 205 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  RR+ )
1713, 16rerpdivcld 10412 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  e.  RR )
182adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  RR )
1918, 16rerpdivcld 10412 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  e.  RR )
204adantr 453 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  A )
213adantr 453 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  RR )
22 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <  ( A  +  B ) )
23 divge0 9620 . . . . . . . 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 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  RR )
2617, 24, 25recxpcld 20064 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  e.  RR )
275adantr 453 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  B )
28 divge0 9620 . . . . . . . 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 20064 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  e.  RR )
311, 2addge01d 9355 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( A  +  B ) ) )
325, 31mpbid 203 . . . . . . . . . 10  |-  ( ph  ->  A  <_  ( A  +  B ) )
3332adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  <_  ( A  +  B ) )
3421recnd 8856 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  CC )
3534mulid1d 8847 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  x.  1 )  =  ( A  +  B ) )
3633, 35breqtrrd 4050 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  <_  ( ( A  +  B
)  x.  1 ) )
37 1re 8832 . . . . . . . . . 10  |-  1  e.  RR
3837a1i 12 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  e.  RR )
39 ledivmul 9624 . . . . . . . . 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 225 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  <_ 
1 )
427adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  RR+ )
43 cxpaddle.6 . . . . . . . 8  |-  ( ph  ->  C  <_  1 )
4443adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  <_  1 )
4517, 24, 41, 42, 44cxpaddlelem 20085 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  <_ 
( ( A  / 
( A  +  B
) )  ^ c  C ) )
462, 1addge02d 9356 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  <_  A  <->  B  <_  ( A  +  B ) ) )
474, 46mpbid 203 . . . . . . . . . 10  |-  ( ph  ->  B  <_  ( A  +  B ) )
4847adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  <_  ( A  +  B ) )
4948, 35breqtrrd 4050 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  <_  ( ( A  +  B
)  x.  1 ) )
50 ledivmul 9624 . . . . . . . . 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 225 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  <_ 
1 )
5319, 29, 52, 42, 44cxpaddlelem 20085 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  <_ 
( ( B  / 
( A  +  B
) )  ^ c  C ) )
5417, 19, 26, 30, 45, 53le2addd 9385 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  <_  (
( ( A  / 
( A  +  B
) )  ^ c  C )  +  ( ( B  /  ( A  +  B )
)  ^ c  C
) ) )
5513recnd 8856 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  e.  CC )
5618recnd 8856 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  CC )
5716rpne0d 10390 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  =/=  0
)
5855, 56, 34, 57divdird 9569 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  ( ( A  / 
( A  +  B
) )  +  ( B  /  ( A  +  B ) ) ) )
5934, 57dividd 9529 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  1 )
6058, 59eqtr3d 2318 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  =  1 )
618recnd 8856 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
6261adantr 453 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  CC )
6313, 20, 16, 62divcxpd 20063 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  =  ( ( A  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6418, 27, 16, 62divcxpd 20063 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  =  ( ( B  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6563, 64oveq12d 5837 . . . . . 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 20064 . . . . . . . . 9  |-  ( ph  ->  ( A  ^ c  C )  e.  RR )
6766recnd 8856 . . . . . . . 8  |-  ( ph  ->  ( A  ^ c  C )  e.  CC )
6867adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  ^ c  C )  e.  CC )
692, 5, 8recxpcld 20064 . . . . . . . . 9  |-  ( ph  ->  ( B  ^ c  C )  e.  RR )
7069recnd 8856 . . . . . . . 8  |-  ( ph  ->  ( B  ^ c  C )  e.  CC )
7170adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  ^ c  C )  e.  CC )
7216, 25rpcxpcld 20071 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR+ )
7372rpne0d 10390 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  =/=  0 )
7468, 71, 11, 73divdird 9569 . . . . . 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 2319 . . . . 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 4053 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  <_  ( ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  /  (
( A  +  B
)  ^ c  C
) ) )
7766, 69readdcld 8857 . . . . . 6  |-  ( ph  ->  ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  e.  RR )
7877adantr 453 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  ^ c  C )  +  ( B  ^ c  C ) )  e.  RR )
7938, 78, 72lemuldivd 10430 . . . 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 225 . . 3  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( 1  x.  ( ( A  +  B )  ^ c  C ) )  <_ 
( ( A  ^ c  C )  +  ( B  ^ c  C
) ) )
8112, 80eqbrtrrd 4046 . 2  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C
) ) )
827rpne0d 10390 . . . . . 6  |-  ( ph  ->  C  =/=  0 )
8361, 820cxpd 20051 . . . . 5  |-  ( ph  ->  ( 0  ^ c  C )  =  0 )
841, 4, 8cxpge0d 20065 . . . . . 6  |-  ( ph  ->  0  <_  ( A  ^ c  C )
)
852, 5, 8cxpge0d 20065 . . . . . 6  |-  ( ph  ->  0  <_  ( B  ^ c  C )
)
8666, 69, 84, 85addge0d 9343 . . . . 5  |-  ( ph  ->  0  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) ) )
8783, 86eqbrtrd 4044 . . . 4  |-  ( ph  ->  ( 0  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) )
88 oveq1 5826 . . . . 5  |-  ( 0  =  ( A  +  B )  ->  (
0  ^ c  C
)  =  ( ( A  +  B )  ^ c  C ) )
8988breq1d 4034 . . . 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 213 . . 3  |-  ( ph  ->  ( 0  =  ( A  +  B )  ->  ( ( A  +  B )  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) ) )
9190imp 420 . 2  |-  ( (
ph  /\  0  =  ( A  +  B
) )  ->  (
( A  +  B
)  ^ c  C
)  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) ) )
92 0re 8833 . . . 4  |-  0  e.  RR
93 leloe 8903 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  +  B
)  e.  RR )  ->  ( 0  <_ 
( A  +  B
)  <->  ( 0  < 
( A  +  B
)  \/  0  =  ( A  +  B
) ) ) )
9492, 3, 93sylancr 646 . . 3  |-  ( ph  ->  ( 0  <_  ( A  +  B )  <->  ( 0  <  ( A  +  B )  \/  0  =  ( A  +  B ) ) ) )
956, 94mpbid 203 . 2  |-  ( ph  ->  ( 0  <  ( A  +  B )  \/  0  =  ( A  +  B )
) )
9681, 91, 95mpjaodan 763 1  |-  ( ph  ->  ( ( A  +  B )  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   class class class wbr 4024  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    + caddc 8735    x. cmul 8737    < clt 8862    <_ cle 8863    / cdiv 9418   RR+crp 10349    ^ c ccxp 19907
This theorem is referenced by:  abvcxp  20758
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ioc 10655  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-mod 10968  df-seq 11041  df-exp 11099  df-fac 11283  df-bc 11310  df-hash 11332  df-shft 11556  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-limsup 11939  df-clim 11956  df-rlim 11957  df-sum 12153  df-ef 12343  df-sin 12345  df-cos 12346  df-pi 12348  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cncf 18376  df-limc 19210  df-dv 19211  df-log 19908  df-cxp 19909
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