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Theorem padicabvcxp 20729
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 20714 . . . . . 6  |-  ( ( P  e.  Prime  /\  y  e.  QQ )  ->  (
( J `  P
) `  y )  =  if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) ) )
32adantlr 698 . . . . 5  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( ( J `
 P ) `  y )  =  if ( y  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  y ) ) ) )
43oveq1d 5793 . . . 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 5785 . . . . . . 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 5785 . . . . . . 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 3534 . . . . . 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 10313 . . . . . . . . . . 11  |-  ( R  e.  RR+  ->  R  e.  RR )
98adantl 454 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  RR )
109recnd 8815 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  e.  CC )
11 rpne0 10322 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  =/=  0 )
1211adantl 454 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  R  =/=  0 )
1310, 120cxpd 20005 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
0  ^ c  R
)  =  0 )
1413adantr 453 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  y  e.  QQ )  ->  ( 0  ^ c  R )  =  0 )
1514ifeq1d 3539 . . . . . 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 2300 . . . . 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 2421 . . . . . . 7  |-  ( y  =/=  0  <->  -.  y  =  0 )
18 pcqcl 12857 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
1918adantlr 698 . . . . . . . . . . . . . 14  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
2019zcnd 10071 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  CC )
2110adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  R  e.  CC )
22 mulneg12 9172 . . . . . . . . . . . . 13  |-  ( ( ( P  pCnt  y
)  e.  CC  /\  R  e.  CC )  ->  ( -u ( P 
pCnt  y )  x.  R )  =  ( ( P  pCnt  y
)  x.  -u R
) )
2320, 21, 22syl2anc 645 . . . . . . . . . . . 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 9098 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  CC )
2520, 24mulcomd 8810 . . . . . . . . . . . 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 2288 . . . . . . . . . . 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 5794 . . . . . . . . . 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 12724 . . . . . . . . . . . . . . . 16  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2928adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  ( ZZ>= `  2 )
)
30 eluz2b2 10243 . . . . . . . . . . . . . . 15  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
3129, 30sylib 190 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  e.  NN  /\  1  <  P ) )
3231simpld 447 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  NN )
3332nnrpd 10342 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR+ )
3433adantr 453 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  e.  RR+ )
3519znegcld 10072 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  ZZ )
3635zred 10070 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u ( P 
pCnt  y )  e.  RR )
3734, 36, 21cxpmuld 20029 . . . . . . . . . 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 9164 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  -u R  e.  RR )
3938adantr 453 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  -u R  e.  RR )
4034, 39, 20cxpmuld 20029 . . . . . . . . . 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 2296 . . . . . . . . 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 9715 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  RR )
4342recnd 8815 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  e.  CC )
4443adantr 453 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  e.  CC )
4532nnne0d 9744 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  P  =/=  0 )
4645adantr 453 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  P  =/=  0 )
4744, 46, 35cxpexpzd 20006 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u ( P 
pCnt  y ) )  =  ( P ^ -u ( P  pCnt  y
) ) )
4847oveq1d 5793 . . . . . . . . 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 20025 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  e.  RR+ )
5049adantr 453 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u R )  e.  RR+ )
5150rpcnd 10345 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u R )  e.  CC )
52 rpne0 10322 . . . . . . . . . . 11  |-  ( ( P  ^ c  -u R )  e.  RR+  ->  ( P  ^ c  -u R )  =/=  0
)
5350, 52syl 17 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  R  e.  RR+ )  /\  ( y  e.  QQ  /\  y  =/=  0 ) )  ->  ( P  ^ c  -u R )  =/=  0 )
5451, 53, 19cxpexpzd 20006 . . . . . . . . 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 2296 . . . . . . . 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 632 . . . . . . 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 464 . . . . . 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 3551 . . . . 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 2288 . . . 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 2288 . . 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 4066 . 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 10313 . . . . 5  |-  ( ( P  ^ c  -u R )  e.  RR+  ->  ( P  ^ c  -u R )  e.  RR )
6349, 62syl 17 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  e.  RR )
64 rpgt0 10318 . . . . 5  |-  ( ( P  ^ c  -u R )  e.  RR+  ->  0  <  ( P  ^ c  -u R
) )
6549, 64syl 17 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  ( P  ^ c  -u R ) )
66 rpgt0 10318 . . . . . . . 8  |-  ( R  e.  RR+  ->  0  < 
R )
6766adantl 454 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  <  R )
689lt0neg2d 9297 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  (
0  <  R  <->  -u R  <  0 ) )
6967, 68mpbid 203 . . . . . 6  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  -u R  <  0 )
7031simprd 451 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  1  <  P )
71 0re 8792 . . . . . . . 8  |-  0  e.  RR
7271a1i 12 . . . . . . 7  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  0  e.  RR )
7342, 70, 38, 72cxpltd 20014 . . . . . 6  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( -u R  <  0  <->  ( P  ^ c  -u R
)  <  ( P  ^ c  0 ) ) )
7469, 73mpbid 203 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  <  ( P  ^ c  0 ) )
7543cxp0d 20000 . . . . 5  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  0 )  =  1 )
7674, 75breqtrd 4007 . . . 4  |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( P  ^ c  -u R
)  <  1 )
77 0xr 8832 . . . . 5  |-  0  e.  RR*
78 ressxr 8830 . . . . . 6  |-  RR  C_  RR*
79 1re 8791 . . . . . 6  |-  1  e.  RR
8078, 79sselii 3138 . . . . 5  |-  1  e.  RR*
81 elioo2 10649 . . . . 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 656 . . . 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 1141 . . 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 2256 . . . 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 20727 . . 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 458 . 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 2330 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   ifcif 3525   class class class wbr 3983    e. cmpt 4037   ` cfv 4659  (class class class)co 5778   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    x. cmul 8696   RR*cxr 8820    < clt 8821   -ucneg 8992   NNcn 9700   2c2 9749   ZZcz 9977   ZZ>=cuz 10183   QQcq 10269   RR+crp 10307   (,)cioo 10608   ^cexp 11056   Primecprime 12706    pCnt cpc 12837   ↾s cress 13097  AbsValcabv 15529  ℂfldccnfld 16325    ^ c ccxp 19861
This theorem is referenced by:  ostth3  20735  ostth  20736
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-tpos 6154  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-divides 12480  df-gcd 12634  df-prime 12707  df-pc 12838  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-grp 14437  df-minusg 14438  df-mulg 14440  df-subg 14566  df-cntz 14741  df-cmn 15039  df-mgp 15274  df-ring 15288  df-cring 15289  df-ur 15290  df-oppr 15353  df-dvdsr 15371  df-unit 15372  df-invr 15402  df-dvr 15413  df-drng 15462  df-subrg 15491  df-abv 15530  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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