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Theorem padicabvcxp 22881
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 22866 . . . . . 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 6106 . . . 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 6098 . . . . . . 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 6098 . . . . . . 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 3802 . . . . . 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 10997 . . . . . . . . . . 11  |-  ( R  e.  RR+  ->  R  e.  RR )
98adantl 466 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  RR )
109recnd 9412 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  CC )
11 rpne0 11006 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  =/=  0 )
1211adantl 466 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  =/=  0 )
1310, 120cxpd 22155 . . . . . . . 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 3807 . . . . . 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 2487 . . . . 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 2608 . . . . . . 7  |-  ( y  =/=  0  <->  -.  y  =  0 )
18 pcqcl 13923 . . . . . . . . . . . . . . 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 10748 . . . . . . . . . . . . 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 9783 . . . . . . . . . . . . 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 9706 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  CC )
2520, 24mulcomd 9407 . . . . . . . . . . . 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 2475 . . . . . . . . . . 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 6107 . . . . . . . . . 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 13781 . . . . . . . . . . . . . . . 16  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2928adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  ( ZZ>= `  2 )
)
30 eluz2b2 10927 . . . . . . . . . . . . . . 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 11026 . . . . . . . . . . . 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 10749 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  ZZ )
3635zred 10747 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  RR )
3734, 36, 21cxpmuld 22179 . . . . . . . . . 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 9775 . . . . . . . . . . . 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 22179 . . . . . . . . . 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 2483 . . . . . . . . 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 10337 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR )
4342recnd 9412 . . . . . . . . . . . 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 10366 . . . . . . . . . . . 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 22156 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^c  -u ( P 
pCnt  y ) )  =  ( P ^ -u ( P  pCnt  y
) ) )
4847oveq1d 6106 . . . . . . . . 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 22175 . . . . . . . . . . . 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 11029 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^c  -u R )  e.  CC )
52 rpne0 11006 . . . . . . . . . . 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 22156 . . . . . . . . 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 2483 . . . . . . . 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 3820 . . . . 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 2475 . . . 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 2475 . . 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 4378 . 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 10997 . . . . 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 11002 . . . . 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 11002 . . . . . . . 8  |-  ( R  e.  RR+  ->  0  < 
R )
6766adantl 466 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  R )
689lt0neg2d 9910 . . . . . . 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 9387 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  e.  RR )
7242, 70, 38, 71cxpltd 22164 . . . . . 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 22150 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  0 )  =  1 )
7573, 74breqtrd 4316 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^c  -u R
)  <  1 )
76 0xr 9430 . . . . 5  |-  0  e.  RR*
77 1re 9385 . . . . . 6  |-  1  e.  RR
7877rexri 9436 . . . . 5  |-  1  e.  RR*
79 elioo2 11341 . . . . 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 1172 . . 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 2443 . . . 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 22879 . . 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 2517 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   ifcif 3791   class class class wbr 4292    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    x. cmul 9287   RR*cxr 9417    < clt 9418   -ucneg 9596   NNcn 10322   2c2 10371   ZZcz 10646   ZZ>=cuz 10861   QQcq 10953   RR+crp 10991   (,)cioo 11300   ^cexp 11865   Primecprime 13763    pCnt cpc 13903   ↾s cress 14175  AbsValcabv 16901  ℂfldccnfld 17818    ^c ccxp 22007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-dvds 13536  df-gcd 13691  df-prm 13764  df-pc 13904  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-mulg 15548  df-subg 15678  df-cntz 15835  df-cmn 16279  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-subrg 16863  df-abv 16902  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-cxp 22009
This theorem is referenced by:  ostth3  22887  ostth  22888
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