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Theorem cxpaddle 20636
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 9115 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  RR )
4 cxpaddle.2 . . . . . . . 8  |-  ( ph  ->  0  <_  A )
5 cxpaddle.4 . . . . . . . 8  |-  ( ph  ->  0  <_  B )
61, 2, 4, 5addge0d 9602 . . . . . . 7  |-  ( ph  ->  0  <_  ( A  +  B ) )
7 cxpaddle.5 . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
87rpred 10648 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
93, 6, 8recxpcld 20614 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
109adantr 452 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
1110recnd 9114 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  CC )
1211mulid2d 9106 . . 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 10614 . . . . . . . 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 10675 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  e.  RR )
182adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  RR )
1918, 16rerpdivcld 10675 . . . . . 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 9879 . . . . . . . 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 20614 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  e.  RR )
275adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  B )
28 divge0 9879 . . . . . . . 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 20614 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  e.  RR )
311, 2addge01d 9614 . . . . . . . . . . 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 9114 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  CC )
3534mulid1d 9105 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  x.  1 )  =  ( A  +  B ) )
3633, 35breqtrrd 4238 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  <_  ( ( A  +  B
)  x.  1 ) )
37 1re 9090 . . . . . . . . . 10  |-  1  e.  RR
3837a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  e.  RR )
39 ledivmul 9883 . . . . . . . . 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 20635 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  <_ 
( ( A  / 
( A  +  B
) )  ^ c  C ) )
462, 1addge02d 9615 . . . . . . . . . . 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 4238 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  <_  ( ( A  +  B
)  x.  1 ) )
50 ledivmul 9883 . . . . . . . . 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 20635 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  <_ 
( ( B  / 
( A  +  B
) )  ^ c  C ) )
5417, 19, 26, 30, 45, 53le2addd 9644 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  <_  (
( ( A  / 
( A  +  B
) )  ^ c  C )  +  ( ( B  /  ( A  +  B )
)  ^ c  C
) ) )
5513recnd 9114 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  e.  CC )
5618recnd 9114 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  CC )
5716rpne0d 10653 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  =/=  0
)
5855, 56, 34, 57divdird 9828 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  ( ( A  / 
( A  +  B
) )  +  ( B  /  ( A  +  B ) ) ) )
5934, 57dividd 9788 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  1 )
6058, 59eqtr3d 2470 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  =  1 )
618recnd 9114 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
6261adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  CC )
6313, 20, 16, 62divcxpd 20613 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  =  ( ( A  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6418, 27, 16, 62divcxpd 20613 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  =  ( ( B  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6563, 64oveq12d 6099 . . . . . 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 20614 . . . . . . . . 9  |-  ( ph  ->  ( A  ^ c  C )  e.  RR )
6766recnd 9114 . . . . . . . 8  |-  ( ph  ->  ( A  ^ c  C )  e.  CC )
6867adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  ^ c  C )  e.  CC )
692, 5, 8recxpcld 20614 . . . . . . . . 9  |-  ( ph  ->  ( B  ^ c  C )  e.  RR )
7069recnd 9114 . . . . . . . 8  |-  ( ph  ->  ( B  ^ c  C )  e.  CC )
7170adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  ^ c  C )  e.  CC )
7216, 25rpcxpcld 20621 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR+ )
7372rpne0d 10653 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  =/=  0 )
7468, 71, 11, 73divdird 9828 . . . . . 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 2471 . . . . 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 4241 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  <_  ( ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  /  (
( A  +  B
)  ^ c  C
) ) )
7766, 69readdcld 9115 . . . . . 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 10693 . . . 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 4234 . 2  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C
) ) )
827rpne0d 10653 . . . . . 6  |-  ( ph  ->  C  =/=  0 )
8361, 820cxpd 20601 . . . . 5  |-  ( ph  ->  ( 0  ^ c  C )  =  0 )
841, 4, 8cxpge0d 20615 . . . . . 6  |-  ( ph  ->  0  <_  ( A  ^ c  C )
)
852, 5, 8cxpge0d 20615 . . . . . 6  |-  ( ph  ->  0  <_  ( B  ^ c  C )
)
8666, 69, 84, 85addge0d 9602 . . . . 5  |-  ( ph  ->  0  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) ) )
8783, 86eqbrtrd 4232 . . . 4  |-  ( ph  ->  ( 0  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) )
88 oveq1 6088 . . . . 5  |-  ( 0  =  ( A  +  B )  ->  (
0  ^ c  C
)  =  ( ( A  +  B )  ^ c  C ) )
8988breq1d 4222 . . . 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 9091 . . . 4  |-  0  e.  RR
93 leloe 9161 . . . 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 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    / cdiv 9677   RR+crp 10612    ^ c ccxp 20453
This theorem is referenced by:  abvcxp  21309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-pi 12675  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454  df-cxp 20455
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