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Theorem cxpaddle 20040
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 8816 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  RR )
4 cxpaddle.2 . . . . . . . 8  |-  ( ph  ->  0  <_  A )
5 cxpaddle.4 . . . . . . . 8  |-  ( ph  ->  0  <_  B )
61, 2, 4, 5addge0d 9302 . . . . . . 7  |-  ( ph  ->  0  <_  ( A  +  B ) )
7 cxpaddle.5 . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
87rpred 10343 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
93, 6, 8recxpcld 20018 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
109adantr 453 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR )
1110recnd 8815 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  CC )
1211mulid2d 8807 . . 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 554 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
15 elrp 10309 . . . . . . . 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 10370 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  e.  RR )
182adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  RR )
1918, 16rerpdivcld 10370 . . . . . 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 9579 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( ( A  +  B )  e.  RR  /\  0  <  ( A  +  B ) ) )  ->  0  <_  ( A  /  ( A  +  B ) ) )
2413, 20, 21, 22, 23syl22anc 1188 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  ( A  /  ( A  +  B ) ) )
258adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  RR )
2617, 24, 25recxpcld 20018 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  e.  RR )
275adantr 453 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  B )
28 divge0 9579 . . . . . . . 8  |-  ( ( ( B  e.  RR  /\  0  <_  B )  /\  ( ( A  +  B )  e.  RR  /\  0  <  ( A  +  B ) ) )  ->  0  <_  ( B  /  ( A  +  B ) ) )
2918, 27, 21, 22, 28syl22anc 1188 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  0  <_  ( B  /  ( A  +  B ) ) )
3019, 29, 25recxpcld 20018 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  e.  RR )
311, 2addge01d 9314 . . . . . . . . . . 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 8815 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  e.  CC )
3534mulid1d 8806 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  x.  1 )  =  ( A  +  B ) )
3633, 35breqtrrd 4009 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  <_  ( ( A  +  B
)  x.  1 ) )
37 1re 8791 . . . . . . . . . 10  |-  1  e.  RR
3837a1i 12 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  e.  RR )
39 ledivmul 9583 . . . . . . . . 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 1191 . . . . . . . 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 20039 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  /  ( A  +  B ) )  <_ 
( ( A  / 
( A  +  B
) )  ^ c  C ) )
462, 1addge02d 9315 . . . . . . . . . . 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 4009 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  <_  ( ( A  +  B
)  x.  1 ) )
50 ledivmul 9583 . . . . . . . . 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 1191 . . . . . . . 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 20039 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  /  ( A  +  B ) )  <_ 
( ( B  / 
( A  +  B
) )  ^ c  C ) )
5417, 19, 26, 30, 45, 53le2addd 9344 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  <_  (
( ( A  / 
( A  +  B
) )  ^ c  C )  +  ( ( B  /  ( A  +  B )
)  ^ c  C
) ) )
5513recnd 8815 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  A  e.  CC )
5618recnd 8815 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  B  e.  CC )
5716rpne0d 10348 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  +  B )  =/=  0
)
5855, 56, 34, 57divdird 9528 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  ( ( A  / 
( A  +  B
) )  +  ( B  /  ( A  +  B ) ) ) )
5934, 57dividd 9488 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  /  ( A  +  B ) )  =  1 )
6058, 59eqtr3d 2290 . . . . 5  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  +  ( B  /  ( A  +  B )
) )  =  1 )
618recnd 8815 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
6261adantr 453 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  C  e.  CC )
6313, 20, 16, 62divcxpd 20017 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  /  ( A  +  B ) )  ^ c  C )  =  ( ( A  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6418, 27, 16, 62divcxpd 20017 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( B  /  ( A  +  B ) )  ^ c  C )  =  ( ( B  ^ c  C )  /  (
( A  +  B
)  ^ c  C
) ) )
6563, 64oveq12d 5796 . . . . . 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 20018 . . . . . . . . 9  |-  ( ph  ->  ( A  ^ c  C )  e.  RR )
6766recnd 8815 . . . . . . . 8  |-  ( ph  ->  ( A  ^ c  C )  e.  CC )
6867adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( A  ^ c  C )  e.  CC )
692, 5, 8recxpcld 20018 . . . . . . . . 9  |-  ( ph  ->  ( B  ^ c  C )  e.  RR )
7069recnd 8815 . . . . . . . 8  |-  ( ph  ->  ( B  ^ c  C )  e.  CC )
7170adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( B  ^ c  C )  e.  CC )
7216, 25rpcxpcld 20025 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  e.  RR+ )
7372rpne0d 10348 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  =/=  0 )
7468, 71, 11, 73divdird 9528 . . . . . 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 2291 . . . . 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 4012 . . . 4  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  1  <_  ( ( ( A  ^ c  C )  +  ( B  ^ c  C
) )  /  (
( A  +  B
)  ^ c  C
) ) )
7766, 69readdcld 8816 . . . . . 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 10388 . . . 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 4005 . 2  |-  ( (
ph  /\  0  <  ( A  +  B ) )  ->  ( ( A  +  B )  ^ c  C )  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C
) ) )
827rpne0d 10348 . . . . . 6  |-  ( ph  ->  C  =/=  0 )
8361, 820cxpd 20005 . . . . 5  |-  ( ph  ->  ( 0  ^ c  C )  =  0 )
841, 4, 8cxpge0d 20019 . . . . . 6  |-  ( ph  ->  0  <_  ( A  ^ c  C )
)
852, 5, 8cxpge0d 20019 . . . . . 6  |-  ( ph  ->  0  <_  ( B  ^ c  C )
)
8666, 69, 84, 85addge0d 9302 . . . . 5  |-  ( ph  ->  0  <_  ( ( A  ^ c  C )  +  ( B  ^ c  C ) ) )
8783, 86eqbrtrd 4003 . . . 4  |-  ( ph  ->  ( 0  ^ c  C )  <_  (
( A  ^ c  C )  +  ( B  ^ c  C
) ) )
88 oveq1 5785 . . . . 5  |-  ( 0  =  ( A  +  B )  ->  (
0  ^ c  C
)  =  ( ( A  +  B )  ^ c  C ) )
8988breq1d 3993 . . . 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 8792 . . . 4  |-  0  e.  RR
93 leloe 8862 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  +  B
)  e.  RR )  ->  ( 0  <_ 
( A  +  B
)  <->  ( 0  < 
( A  +  B
)  \/  0  =  ( A  +  B
) ) ) )
9492, 3, 93sylancr 647 . . 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 764 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 1619    e. wcel 1621   class class class wbr 3983  (class class class)co 5778   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    < clt 8821    <_ cle 8822    / cdiv 9377   RR+crp 10307    ^ c ccxp 19861
This theorem is referenced by:  abvcxp  20712
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862  df-cxp 19863
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