MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulcxp Unicode version

Theorem mulcxp 20032
Description: Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
mulcxp  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) )

Proof of Theorem mulcxp
StepHypRef Expression
1 simp1l 979 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  A  e.  RR )
21recnd 8861 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  A  e.  CC )
32mul01d 9011 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  x.  0 )  =  0 )
43oveq1d 5873 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( A  x.  0 )  ^ c  C )  =  ( 0  ^ c  C
) )
5 simp3 957 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  C  e.  CC )
62, 5mulcxplem 20031 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  ^ c  C )  =  ( ( A  ^ c  C )  x.  (
0  ^ c  C
) ) )
74, 6eqtrd 2315 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( A  x.  0 )  ^ c  C )  =  ( ( A  ^ c  C )  x.  (
0  ^ c  C
) ) )
8 oveq2 5866 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
98oveq1d 5873 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( A  x.  0 )  ^ c  C ) )
10 oveq1 5865 . . . . 5  |-  ( B  =  0  ->  ( B  ^ c  C )  =  ( 0  ^ c  C ) )
1110oveq2d 5874 . . . 4  |-  ( B  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( A  ^ c  C
)  x.  ( 0  ^ c  C ) ) )
129, 11eqeq12d 2297 . . 3  |-  ( B  =  0  ->  (
( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) )  <->  ( ( A  x.  0 )  ^ c  C )  =  ( ( A  ^ c  C )  x.  (
0  ^ c  C
) ) ) )
137, 12syl5ibrcom 213 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( B  =  0  ->  ( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) ) )
14 simp2l 981 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  B  e.  RR )
1514recnd 8861 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  B  e.  CC )
1615mul02d 9010 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  x.  B )  =  0 )
1716oveq1d 5873 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( 0  x.  B )  ^ c  C )  =  ( 0  ^ c  C
) )
1815, 5mulcxplem 20031 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  ^ c  C )  =  ( ( B  ^ c  C )  x.  (
0  ^ c  C
) ) )
19 cxpcl 20021 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  ^ c  C )  e.  CC )
2015, 5, 19syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( B  ^ c  C )  e.  CC )
21 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
22 cxpcl 20021 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  C  e.  CC )  ->  ( 0  ^ c  C )  e.  CC )
2321, 5, 22sylancr 644 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  ^ c  C )  e.  CC )
2420, 23mulcomd 8856 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( B  ^ c  C )  x.  ( 0  ^ c  C ) )  =  ( ( 0  ^ c  C )  x.  ( B  ^ c  C ) ) )
2518, 24eqtrd 2315 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  ^ c  C )  =  ( ( 0  ^ c  C )  x.  ( B  ^ c  C ) ) )
2617, 25eqtrd 2315 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( 0  x.  B )  ^ c  C )  =  ( ( 0  ^ c  C )  x.  ( B  ^ c  C ) ) )
27 oveq1 5865 . . . . . . 7  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
2827oveq1d 5873 . . . . . 6  |-  ( A  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( 0  x.  B )  ^ c  C ) )
29 oveq1 5865 . . . . . . 7  |-  ( A  =  0  ->  ( A  ^ c  C )  =  ( 0  ^ c  C ) )
3029oveq1d 5873 . . . . . 6  |-  ( A  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( 0  ^ c  C
)  x.  ( B  ^ c  C ) ) )
3128, 30eqeq12d 2297 . . . . 5  |-  ( A  =  0  ->  (
( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) )  <->  ( ( 0  x.  B )  ^ c  C )  =  ( ( 0  ^ c  C )  x.  ( B  ^ c  C ) ) ) )
3226, 31syl5ibrcom 213 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  =  0  ->  ( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) ) )
3332a1dd 42 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  =  0  ->  ( B  =/=  0  ->  ( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) ) ) )
341adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  e.  RR )
35 simpl1r 1007 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <_  A )
36 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  =/=  0 )
3734, 35, 36ne0gt0d 8956 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  A )
3834, 37elrpd 10388 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  e.  RR+ )
3914adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  RR )
40 simpl2r 1009 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <_  B )
41 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  =/=  0 )
4239, 40, 41ne0gt0d 8956 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  B )
4339, 42elrpd 10388 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  RR+ )
4438, 43relogmuld 19976 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( log `  ( A  x.  B )
)  =  ( ( log `  A )  +  ( log `  B
) ) )
4544oveq2d 5874 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( C  x.  ( log `  ( A  x.  B ) ) )  =  ( C  x.  ( ( log `  A
)  +  ( log `  B ) ) ) )
465adantr 451 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  C  e.  CC )
472adantr 451 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  e.  CC )
48 logcl 19926 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
4947, 36, 48syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( log `  A
)  e.  CC )
5015adantr 451 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  CC )
51 logcl 19926 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( log `  B
)  e.  CC )
5250, 41, 51syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( log `  B
)  e.  CC )
5346, 49, 52adddid 8859 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( C  x.  (
( log `  A
)  +  ( log `  B ) ) )  =  ( ( C  x.  ( log `  A
) )  +  ( C  x.  ( log `  B ) ) ) )
5445, 53eqtrd 2315 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( C  x.  ( log `  ( A  x.  B ) ) )  =  ( ( C  x.  ( log `  A
) )  +  ( C  x.  ( log `  B ) ) ) )
5554fveq2d 5529 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( exp `  ( C  x.  ( log `  ( A  x.  B
) ) ) )  =  ( exp `  (
( C  x.  ( log `  A ) )  +  ( C  x.  ( log `  B ) ) ) ) )
5646, 49mulcld 8855 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( C  x.  ( log `  A ) )  e.  CC )
5746, 52mulcld 8855 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( C  x.  ( log `  B ) )  e.  CC )
58 efadd 12375 . . . . . . 7  |-  ( ( ( C  x.  ( log `  A ) )  e.  CC  /\  ( C  x.  ( log `  B ) )  e.  CC )  ->  ( exp `  ( ( C  x.  ( log `  A
) )  +  ( C  x.  ( log `  B ) ) ) )  =  ( ( exp `  ( C  x.  ( log `  A
) ) )  x.  ( exp `  ( C  x.  ( log `  B ) ) ) ) )
5956, 57, 58syl2anc 642 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( exp `  (
( C  x.  ( log `  A ) )  +  ( C  x.  ( log `  B ) ) ) )  =  ( ( exp `  ( C  x.  ( log `  A ) ) )  x.  ( exp `  ( C  x.  ( log `  B ) ) ) ) )
6055, 59eqtrd 2315 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( exp `  ( C  x.  ( log `  ( A  x.  B
) ) ) )  =  ( ( exp `  ( C  x.  ( log `  A ) ) )  x.  ( exp `  ( C  x.  ( log `  B ) ) ) ) )
6147, 50mulcld 8855 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  CC )
6247, 50, 36, 41mulne0d 9420 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
63 cxpef 20012 . . . . . 6  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( A  x.  B
)  =/=  0  /\  C  e.  CC )  ->  ( ( A  x.  B )  ^ c  C )  =  ( exp `  ( C  x.  ( log `  ( A  x.  B )
) ) ) )
6461, 62, 46, 63syl3anc 1182 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  ^ c  C )  =  ( exp `  ( C  x.  ( log `  ( A  x.  B )
) ) ) )
65 cxpef 20012 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  C  e.  CC )  ->  ( A  ^ c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
6647, 36, 46, 65syl3anc 1182 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( A  ^ c  C )  =  ( exp `  ( C  x.  ( log `  A
) ) ) )
67 cxpef 20012 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  C  e.  CC )  ->  ( B  ^ c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
6850, 41, 46, 67syl3anc 1182 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( B  ^ c  C )  =  ( exp `  ( C  x.  ( log `  B
) ) ) )
6966, 68oveq12d 5876 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( ( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( exp `  ( C  x.  ( log `  A
) ) )  x.  ( exp `  ( C  x.  ( log `  B ) ) ) ) )
7060, 64, 693eqtr4d 2325 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) )
7170exp32 588 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  =/=  0  ->  ( B  =/=  0  ->  ( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) ) ) )
7233, 71pm2.61dne 2523 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( B  =/=  0  ->  ( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) ) )
7313, 72pm2.61dne 2523 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( A  x.  B )  ^ c  C )  =  ( ( A  ^ c  C )  x.  ( B  ^ c  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868   expce 12343   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  cxprec  20033  divcxp  20034  mulcxpd  20075
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-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-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-mulg 14492  df-cntz 14793  df-cmn 15091  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
  Copyright terms: Public domain W3C validator