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

Theorem mulcxp 19959
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 984 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  A  e.  RR )
21recnd 8794 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  A  e.  CC )
32mul01d 8944 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  x.  0 )  =  0 )
43oveq1d 5772 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( A  x.  0 )  ^ c  C )  =  ( 0  ^ c  C
) )
5 simp3 962 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  C  e.  CC )
62, 5mulcxplem 19958 . . . 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 2288 . . 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 5765 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
98oveq1d 5772 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( A  x.  0 )  ^ c  C ) )
10 oveq1 5764 . . . . 5  |-  ( B  =  0  ->  ( B  ^ c  C )  =  ( 0  ^ c  C ) )
1110oveq2d 5773 . . . 4  |-  ( B  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( A  ^ c  C
)  x.  ( 0  ^ c  C ) ) )
129, 11eqeq12d 2270 . . 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 215 . 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 986 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  B  e.  RR )
1514recnd 8794 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  B  e.  CC )
1615mul02d 8943 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  x.  B )  =  0 )
1716oveq1d 5772 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( 0  x.  B )  ^ c  C )  =  ( 0  ^ c  C
) )
1815, 5mulcxplem 19958 . . . . . . 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 19948 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  ^ c  C )  e.  CC )
2015, 5, 19syl2anc 645 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( B  ^ c  C )  e.  CC )
21 0cn 8764 . . . . . . . . 9  |-  0  e.  CC
22 cxpcl 19948 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  C  e.  CC )  ->  ( 0  ^ c  C )  e.  CC )
2321, 5, 22sylancr 647 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  ^ c  C )  e.  CC )
2420, 23mulcomd 8789 . . . . . . 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 2288 . . . . . 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 2288 . . . . 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 5764 . . . . . . 7  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
2827oveq1d 5772 . . . . . 6  |-  ( A  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( 0  x.  B )  ^ c  C ) )
29 oveq1 5764 . . . . . . 7  |-  ( A  =  0  ->  ( A  ^ c  C )  =  ( 0  ^ c  C ) )
3029oveq1d 5772 . . . . . 6  |-  ( A  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( 0  ^ c  C
)  x.  ( B  ^ c  C ) ) )
3128, 30eqeq12d 2270 . . . . 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 215 . . . 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 44 . . 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 453 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  e.  RR )
35 simpl1r 1012 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <_  A )
36 simprl 735 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  =/=  0 )
3734, 35, 36ne0gt0d 8889 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  A )
3834, 37elrpd 10320 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  e.  RR+ )
3914adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  RR )
40 simpl2r 1014 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <_  B )
41 simprr 736 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  =/=  0 )
4239, 40, 41ne0gt0d 8889 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  B )
4339, 42elrpd 10320 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  RR+ )
4438, 43relogmuld 19903 . . . . . . . . 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 5773 . . . . . . . 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 453 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  C  e.  CC )
472adantr 453 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  A  e.  CC )
48 logcl 19853 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
4947, 36, 48syl2anc 645 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( log `  A
)  e.  CC )
5015adantr 453 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  CC )
51 logcl 19853 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( log `  B
)  e.  CC )
5250, 41, 51syl2anc 645 . . . . . . . . 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 8792 . . . . . . . 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 2288 . . . . . . 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 5427 . . . . . 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 8788 . . . . . . 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 8788 . . . . . . 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 12302 . . . . . . 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 645 . . . . . 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 2288 . . . . 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 8788 . . . . . 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 9353 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
63 cxpef 19939 . . . . . 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 1187 . . . . 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 19939 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  C  e.  CC )  ->  ( A  ^ c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
6647, 36, 46, 65syl3anc 1187 . . . . . 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 19939 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  C  e.  CC )  ->  ( B  ^ c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
6850, 41, 46, 67syl3anc 1187 . . . . . 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 5775 . . . . 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 2298 . . . 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 591 . . 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 2496 . 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 2496 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670    + caddc 8673    x. cmul 8675    <_ cle 8801   expce 12270   logclog 19839    ^ c ccxp 19840
This theorem is referenced by:  cxprec  19960  divcxp  19961  mulcxpd  20002
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-sin 12278  df-cos 12279  df-pi 12281  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841  df-cxp 19842
  Copyright terms: Public domain W3C validator