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Theorem padicabvcxp 20797
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 20782 . . . . . 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 5889 . . . 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 5881 . . . . . . 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 5881 . . . . . . 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 3587 . . . . . 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 10376 . . . . . . . . . . 11  |-  ( R  e.  RR+  ->  R  e.  RR )
98adantl 452 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  RR )
109recnd 8877 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  CC )
11 rpne0 10385 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  =/=  0 )
1211adantl 452 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  =/=  0 )
1310, 120cxpd 20073 . . . . . . . 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 3592 . . . . . 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 2340 . . . . 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 2461 . . . . . . 7  |-  ( y  =/=  0  <->  -.  y  =  0 )
18 pcqcl 12925 . . . . . . . . . . . . . . 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 10134 . . . . . . . . . . . . 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 9234 . . . . . . . . . . . . 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 9160 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  CC )
2520, 24mulcomd 8872 . . . . . . . . . . . 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 2328 . . . . . . . . . . 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 5890 . . . . . . . . . 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 12792 . . . . . . . . . . . . . . . 16  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2928adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  ( ZZ>= `  2 )
)
30 eluz2b2 10306 . . . . . . . . . . . . . . 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 10405 . . . . . . . . . . . 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 10135 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  ZZ )
3635zred 10133 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  RR )
3734, 36, 21cxpmuld 20097 . . . . . . . . . 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 9226 . . . . . . . . . . . 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 20097 . . . . . . . . . 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 2336 . . . . . . . . 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 9777 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR )
4342recnd 8877 . . . . . . . . . . . 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 9806 . . . . . . . . . . . 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 20074 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u ( P 
pCnt  y ) )  =  ( P ^ -u ( P  pCnt  y
) ) )
4847oveq1d 5889 . . . . . . . . 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 20093 . . . . . . . . . . . 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 10408 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u R )  e.  CC )
52 rpne0 10385 . . . . . . . . . . 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 20074 . . . . . . . . 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 2336 . . . . . . . 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 3604 . . . . 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 2328 . . . 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 2328 . . 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 4122 . 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 10376 . . . . 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 10381 . . . . 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 10381 . . . . . . . 8  |-  ( R  e.  RR+  ->  0  < 
R )
6766adantl 452 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  R )
689lt0neg2d 9359 . . . . . . 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 8854 . . . . . . . 8  |-  0  e.  RR
7271a1i 10 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  e.  RR )
7342, 70, 38, 72cxpltd 20082 . . . . . 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 20068 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  0 )  =  1 )
7674, 75breqtrd 4063 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  <  1 )
77 0xr 8894 . . . . 5  |-  0  e.  RR*
78 ressxr 8892 . . . . . 6  |-  RR  C_  RR*
79 1re 8853 . . . . . 6  |-  1  e.  RR
8078, 79sselii 3190 . . . . 5  |-  1  e.  RR*
81 elioo2 10713 . . . . 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 2296 . . . 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 20795 . . 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 2370 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 1632    e. wcel 1696    =/= wne 2459   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758   RR*cxr 8882    < clt 8883   -ucneg 9054   NNcn 9762   2c2 9811   ZZcz 10040   ZZ>=cuz 10246   QQcq 10332   RR+crp 10370   (,)cioo 10672   ^cexp 11120   Primecprime 12774    pCnt cpc 12905   ↾s cress 13165  AbsValcabv 15597  ℂfldccnfld 16393    ^ c ccxp 19929
This theorem is referenced by:  ostth3  20803  ostth  20804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-cntz 14809  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-subrg 15559  df-abv 15598  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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