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Theorem padicabvcxp 23542
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 23527 . . . . . 6  |-  ( ( P  e.  Prime  /\  y  e.  QQ )  ->  (
( J `  P
) `  y )  =  if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) ) )
32adantlr 714 . . . . 5  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( ( J `
 P ) `  y )  =  if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) ) )
43oveq1d 6297 . . . 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 6289 . . . . . . 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 6289 . . . . . . 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 3952 . . . . . 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 11222 . . . . . . . . . . 11  |-  ( R  e.  RR+  ->  R  e.  RR )
98adantl 466 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  RR )
109recnd 9618 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  CC )
11 rpne0 11231 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  =/=  0 )
1211adantl 466 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  =/=  0 )
1310, 120cxpd 22816 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
0  ^c  R )  =  0 )
1413adantr 465 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( 0  ^c  R )  =  0 )
1514ifeq1d 3957 . . . . . 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 2520 . . . . 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 2664 . . . . . . 7  |-  ( y  =/=  0  <->  -.  y  =  0 )
18 pcqcl 14232 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
1918adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
2019zcnd 10963 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  CC )
2110adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  R  e.  CC )
22 mulneg12 9991 . . . . . . . . . . . . 13  |-  ( ( ( P  pCnt  y
)  e.  CC  /\  R  e.  CC )  ->  ( -u ( P 
pCnt  y )  x.  R )  =  ( ( P  pCnt  y
)  x.  -u R
) )
2320, 21, 22syl2anc 661 . . . . . . . . . . . 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 9913 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  CC )
2520, 24mulcomd 9613 . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 6298 . . . . . . . . . 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 14087 . . . . . . . . . . . . . . . 16  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2928adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  ( ZZ>= `  2 )
)
30 eluz2b2 11150 . . . . . . . . . . . . . . 15  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
3129, 30sylib 196 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  e.  NN  /\  1  <  P ) )
3231simpld 459 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  NN )
3332nnrpd 11251 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR+ )
3433adantr 465 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  e.  RR+ )
3519znegcld 10964 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  ZZ )
3635zred 10962 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  RR )
3734, 36, 21cxpmuld 22840 . . . . . . . . . 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 9982 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  -u R  e.  RR )
3938adantr 465 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  RR )
4034, 39, 20cxpmuld 22840 . . . . . . . . . 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 2516 . . . . . . . . 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 10547 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR )
4342recnd 9618 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  CC )
4443adantr 465 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  e.  CC )
4532nnne0d 10576 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  =/=  0 )
4645adantr 465 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  =/=  0 )
4744, 46, 35cxpexpzd 22817 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^c  -u ( P 
pCnt  y ) )  =  ( P ^ -u ( P  pCnt  y
) ) )
4847oveq1d 6297 . . . . . . . . 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 22836 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  -u R
)  e.  RR+ )
5049adantr 465 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^c  -u R )  e.  RR+ )
5150rpcnd 11254 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^c  -u R )  e.  CC )
52 rpne0 11231 . . . . . . . . . . 11  |-  ( ( P  ^c  -u R )  e.  RR+  ->  ( P  ^c  -u R )  =/=  0
)
5350, 52syl 16 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^c  -u R )  =/=  0 )
5451, 53, 19cxpexpzd 22817 . . . . . . . . 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 2516 . . . . . . . 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 648 . . . . . . 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 476 . . . . . 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 3970 . . . . 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 2508 . . . 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 2508 . . 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 4533 . 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 11222 . . . . 5  |-  ( ( P  ^c  -u R )  e.  RR+  ->  ( P  ^c  -u R )  e.  RR )
6349, 62syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  -u R
)  e.  RR )
64 rpgt0 11227 . . . . 5  |-  ( ( P  ^c  -u R )  e.  RR+  ->  0  <  ( P  ^c  -u R
) )
6549, 64syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  ( P  ^c  -u R ) )
66 rpgt0 11227 . . . . . . . 8  |-  ( R  e.  RR+  ->  0  < 
R )
6766adantl 466 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  R )
689lt0neg2d 10119 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
0  <  R  <->  -u R  <  0 ) )
6967, 68mpbid 210 . . . . . 6  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  -u R  <  0 )
7031simprd 463 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  1  <  P )
71 0red 9593 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  e.  RR )
7242, 70, 38, 71cxpltd 22825 . . . . . 6  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( -u R  <  0  <->  ( P  ^c  -u R
)  <  ( P  ^c  0 ) ) )
7369, 72mpbid 210 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  -u R
)  <  ( P  ^c  0 ) )
7443cxp0d 22811 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  0 )  =  1 )
7573, 74breqtrd 4471 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  -u R
)  <  1 )
76 0xr 9636 . . . . 5  |-  0  e.  RR*
77 1re 9591 . . . . . 6  |-  1  e.  RR
7877rexri 9642 . . . . 5  |-  1  e.  RR*
79 elioo2 11566 . . . . 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 ) ) )
8076, 78, 79mp2an 672 . . . 4  |-  ( ( P  ^c  -u R )  e.  ( 0 (,) 1 )  <-> 
( ( P  ^c  -u R )  e.  RR  /\  0  < 
( P  ^c  -u R )  /\  ( P  ^c  -u R
)  <  1 ) )
8163, 65, 75, 80syl3anbrc 1180 . . 3  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  -u R
)  e.  ( 0 (,) 1 ) )
82 qrng.q . . . 4  |-  Q  =  (flds  QQ )
83 qabsabv.a . . . 4  |-  A  =  (AbsVal `  Q )
84 eqid 2467 . . . 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 ) ) ) )
8582, 83, 84padicabv 23540 . . 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
)
8681, 85syldan 470 . 2  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
y  e.  QQ  |->  if ( y  =  0 ,  0 ,  ( ( P  ^c  -u R ) ^ ( P  pCnt  y ) ) ) )  e.  A
)
8761, 86eqeltrd 2555 1  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
y  e.  QQ  |->  ( ( ( J `  P ) `  y
)  ^c  R ) )  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493   RR*cxr 9623    < clt 9624   -ucneg 9802   NNcn 10532   2c2 10581   ZZcz 10860   ZZ>=cuz 11078   QQcq 11178   RR+crp 11216   (,)cioo 11525   ^cexp 12129   Primecprime 14069    pCnt cpc 14212   ↾s cress 14484  AbsValcabv 17245  ℂfldccnfld 18188    ^c ccxp 22668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-fac 12316  df-bc 12343  df-hash 12368  df-shft 12857  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-limsup 13250  df-clim 13267  df-rlim 13268  df-sum 13465  df-ef 13658  df-sin 13660  df-cos 13661  df-pi 13663  df-dvds 13841  df-gcd 13997  df-prm 14070  df-pc 14213  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-mulg 15858  df-subg 15990  df-cntz 16147  df-cmn 16593  df-mgp 16929  df-ur 16941  df-rng 16985  df-cring 16986  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-dvr 17113  df-drng 17178  df-subrg 17207  df-abv 17246  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cncf 21114  df-limc 22002  df-dv 22003  df-log 22669  df-cxp 22670
This theorem is referenced by:  ostth3  23548  ostth  23549
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