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Theorem padicabvcxp 20781
Description: All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
Hypotheses
Ref Expression
qrng.q  |-  Q  =  (flds  QQ )
qabsabv.a  |-  A  =  (AbsVal `  Q )
padic.j  |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( q ^ -u (
q  pCnt  x )
) ) ) )
Assertion
Ref Expression
padicabvcxp  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
y  e.  QQ  |->  ( ( ( J `  P ) `  y
)  ^ c  R
) )  e.  A
)
Distinct variable groups:    x, q,
y    y, J    A, q, x, y    x, Q, y    P, q, x, y    R, q, y
Allowed substitution hints:    Q( q)    R( x)    J( x, q)

Proof of Theorem padicabvcxp
StepHypRef Expression
1 padic.j . . . . . . 7  |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( q ^ -u (
q  pCnt  x )
) ) ) )
21padicval 20766 . . . . . 6  |-  ( ( P  e.  Prime  /\  y  e.  QQ )  ->  (
( J `  P
) `  y )  =  if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) ) )
32adantlr 695 . . . . 5  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( ( J `
 P ) `  y )  =  if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) ) )
43oveq1d 5873 . . . 4  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( ( ( J `  P ) `
 y )  ^ c  R )  =  ( if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) )  ^ c  R
) )
5 oveq1 5865 . . . . . . 7  |-  ( if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) )  =  0  -> 
( if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y
) ) )  ^ c  R )  =  ( 0  ^ c  R
) )
6 oveq1 5865 . . . . . . 7  |-  ( if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) )  =  ( P ^ -u ( P 
pCnt  y ) )  ->  ( if ( y  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  y ) ) )  ^ c  R
)  =  ( ( P ^ -u ( P  pCnt  y ) )  ^ c  R ) )
75, 6ifsb 3574 . . . . . 6  |-  ( if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) )  ^ c  R
)  =  if ( y  =  0 ,  ( 0  ^ c  R ) ,  ( ( P ^ -u ( P  pCnt  y ) )  ^ c  R ) )
8 rpre 10360 . . . . . . . . . . 11  |-  ( R  e.  RR+  ->  R  e.  RR )
98adantl 452 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  RR )
109recnd 8861 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  CC )
11 rpne0 10369 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  =/=  0 )
1211adantl 452 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  =/=  0 )
1310, 120cxpd 20057 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
0  ^ c  R
)  =  0 )
1413adantr 451 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( 0  ^ c  R )  =  0 )
1514ifeq1d 3579 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  if ( y  =  0 ,  ( 0  ^ c  R
) ,  ( ( P ^ -u ( P  pCnt  y ) )  ^ c  R ) )  =  if ( y  =  0 ,  0 ,  ( ( P ^ -u ( P  pCnt  y ) )  ^ c  R ) ) )
167, 15syl5eq 2327 . . . . 5  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( if ( y  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  y ) ) )  ^ c  R
)  =  if ( y  =  0 ,  0 ,  ( ( P ^ -u ( P  pCnt  y ) )  ^ c  R ) ) )
17 df-ne 2448 . . . . . . 7  |-  ( y  =/=  0  <->  -.  y  =  0 )
18 pcqcl 12909 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
1918adantlr 695 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
2019zcnd 10118 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  CC )
2110adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  R  e.  CC )
22 mulneg12 9218 . . . . . . . . . . . . 13  |-  ( ( ( P  pCnt  y
)  e.  CC  /\  R  e.  CC )  ->  ( -u ( P 
pCnt  y )  x.  R )  =  ( ( P  pCnt  y
)  x.  -u R
) )
2320, 21, 22syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( -u ( P  pCnt  y )  x.  R )  =  ( ( P  pCnt  y
)  x.  -u R
) )
2421negcld 9144 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  CC )
2520, 24mulcomd 8856 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( ( P  pCnt  y )  x.  -u R )  =  (
-u R  x.  ( P  pCnt  y ) ) )
2623, 25eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( -u ( P  pCnt  y )  x.  R )  =  (
-u R  x.  ( P  pCnt  y ) ) )
2726oveq2d 5874 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  ( -u ( P  pCnt  y )  x.  R ) )  =  ( P  ^ c 
( -u R  x.  ( P  pCnt  y ) ) ) )
28 prmuz2 12776 . . . . . . . . . . . . . . . 16  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2928adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  ( ZZ>= `  2 )
)
30 eluz2b2 10290 . . . . . . . . . . . . . . 15  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
3129, 30sylib 188 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  e.  NN  /\  1  <  P ) )
3231simpld 445 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  NN )
3332nnrpd 10389 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR+ )
3433adantr 451 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  e.  RR+ )
3519znegcld 10119 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  ZZ )
3635zred 10117 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  RR )
3734, 36, 21cxpmuld 20081 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  ( -u ( P  pCnt  y )  x.  R ) )  =  ( ( P  ^ c  -u ( P  pCnt  y ) )  ^ c  R ) )
389renegcld 9210 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  -u R  e.  RR )
3938adantr 451 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  RR )
4034, 39, 20cxpmuld 20081 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  ( -u R  x.  ( P  pCnt  y
) ) )  =  ( ( P  ^ c  -u R )  ^ c  ( P  pCnt  y ) ) )
4127, 37, 403eqtr3d 2323 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( ( P  ^ c  -u ( P  pCnt  y ) )  ^ c  R )  =  ( ( P  ^ c  -u R
)  ^ c  ( P  pCnt  y )
) )
4232nnred 9761 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR )
4342recnd 8861 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  CC )
4443adantr 451 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  e.  CC )
4532nnne0d 9790 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  =/=  0 )
4645adantr 451 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  =/=  0 )
4744, 46, 35cxpexpzd 20058 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u ( P 
pCnt  y ) )  =  ( P ^ -u ( P  pCnt  y
) ) )
4847oveq1d 5873 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( ( P  ^ c  -u ( P  pCnt  y ) )  ^ c  R )  =  ( ( P ^ -u ( P 
pCnt  y ) )  ^ c  R ) )
4933, 38rpcxpcld 20077 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  e.  RR+ )
5049adantr 451 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u R )  e.  RR+ )
5150rpcnd 10392 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u R )  e.  CC )
52 rpne0 10369 . . . . . . . . . . 11  |-  ( ( P  ^ c  -u R )  e.  RR+  ->  ( P  ^ c  -u R )  =/=  0
)
5350, 52syl 15 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u R )  =/=  0 )
5451, 53, 19cxpexpzd 20058 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( ( P  ^ c  -u R
)  ^ c  ( P  pCnt  y )
)  =  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) )
5541, 48, 543eqtr3d 2323 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( ( P ^ -u ( P 
pCnt  y ) )  ^ c  R )  =  ( ( P  ^ c  -u R
) ^ ( P 
pCnt  y ) ) )
5655anassrs 629 . . . . . . 7  |-  ( ( ( ( P  e. 
Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  /\  y  =/=  0
)  ->  ( ( P ^ -u ( P 
pCnt  y ) )  ^ c  R )  =  ( ( P  ^ c  -u R
) ^ ( P 
pCnt  y ) ) )
5717, 56sylan2br 462 . . . . . 6  |-  ( ( ( ( P  e. 
Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  /\  -.  y  =  0 )  ->  (
( P ^ -u ( P  pCnt  y ) )  ^ c  R )  =  ( ( P  ^ c  -u R
) ^ ( P 
pCnt  y ) ) )
5857ifeq2da 3591 . . . . 5  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  if ( y  =  0 ,  0 ,  ( ( P ^ -u ( P 
pCnt  y ) )  ^ c  R ) )  =  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) ) )
5916, 58eqtrd 2315 . . . 4  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( if ( y  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  y ) ) )  ^ c  R
)  =  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) ) )
604, 59eqtrd 2315 . . 3  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( ( ( J `  P ) `
 y )  ^ c  R )  =  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) ) )
6160mpteq2dva 4106 . 2  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
y  e.  QQ  |->  ( ( ( J `  P ) `  y
)  ^ c  R
) )  =  ( y  e.  QQ  |->  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) ) ) )
62 rpre 10360 . . . . 5  |-  ( ( P  ^ c  -u R )  e.  RR+  ->  ( P  ^ c  -u R )  e.  RR )
6349, 62syl 15 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  e.  RR )
64 rpgt0 10365 . . . . 5  |-  ( ( P  ^ c  -u R )  e.  RR+  ->  0  <  ( P  ^ c  -u R
) )
6549, 64syl 15 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  ( P  ^ c  -u R ) )
66 rpgt0 10365 . . . . . . . 8  |-  ( R  e.  RR+  ->  0  < 
R )
6766adantl 452 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  R )
689lt0neg2d 9343 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
0  <  R  <->  -u R  <  0 ) )
6967, 68mpbid 201 . . . . . 6  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  -u R  <  0 )
7031simprd 449 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  1  <  P )
71 0re 8838 . . . . . . . 8  |-  0  e.  RR
7271a1i 10 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  e.  RR )
7342, 70, 38, 72cxpltd 20066 . . . . . 6  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( -u R  <  0  <->  ( P  ^ c  -u R
)  <  ( P  ^ c  0 ) ) )
7469, 73mpbid 201 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  <  ( P  ^ c  0 ) )
7543cxp0d 20052 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  0 )  =  1 )
7674, 75breqtrd 4047 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  <  1 )
77 0xr 8878 . . . . 5  |-  0  e.  RR*
78 ressxr 8876 . . . . . 6  |-  RR  C_  RR*
79 1re 8837 . . . . . 6  |-  1  e.  RR
8078, 79sselii 3177 . . . . 5  |-  1  e.  RR*
81 elioo2 10697 . . . . 5  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
( P  ^ c  -u R )  e.  ( 0 (,) 1 )  <-> 
( ( P  ^ c  -u R )  e.  RR  /\  0  < 
( P  ^ c  -u R )  /\  ( P  ^ c  -u R
)  <  1 ) ) )
8277, 80, 81mp2an 653 . . . 4  |-  ( ( P  ^ c  -u R )  e.  ( 0 (,) 1 )  <-> 
( ( P  ^ c  -u R )  e.  RR  /\  0  < 
( P  ^ c  -u R )  /\  ( P  ^ c  -u R
)  <  1 ) )
8363, 65, 76, 82syl3anbrc 1136 . . 3  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  e.  ( 0 (,) 1 ) )
84 qrng.q . . . 4  |-  Q  =  (flds  QQ )
85 qabsabv.a . . . 4  |-  A  =  (AbsVal `  Q )
86 eqid 2283 . . . 4  |-  ( y  e.  QQ  |->  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) ) )  =  ( y  e.  QQ  |->  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) ) )
8784, 85, 86padicabv 20779 . . 3  |-  ( ( P  e.  Prime  /\  ( P  ^ c  -u R
)  e.  ( 0 (,) 1 ) )  ->  ( y  e.  QQ  |->  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R
) ^ ( P 
pCnt  y ) ) ) )  e.  A
)
8883, 87syldan 456 . 2  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
y  e.  QQ  |->  if ( y  =  0 ,  0 ,  ( ( P  ^ c  -u R ) ^ ( P  pCnt  y ) ) ) )  e.  A
)
8961, 88eqeltrd 2357 1  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
y  e.  QQ  |->  ( ( ( J `  P ) `  y
)  ^ c  R
) )  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   RR*cxr 8866    < clt 8867   -ucneg 9038   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   QQcq 10316   RR+crp 10354   (,)cioo 10656   ^cexp 11104   Primecprime 12758    pCnt cpc 12889   ↾s cress 13149  AbsValcabv 15581  ℂfldccnfld 16377    ^ c ccxp 19913
This theorem is referenced by:  ostth3  20787  ostth  20788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-cntz 14793  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-abv 15582  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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