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Theorem mulcxp 20034
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 8863 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  A  e.  CC )
32mul01d 9013 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  x.  0 )  =  0 )
43oveq1d 5875 . . . 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 20033 . . . 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 2317 . . 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 5868 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
98oveq1d 5875 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( A  x.  0 )  ^ c  C ) )
10 oveq1 5867 . . . . 5  |-  ( B  =  0  ->  ( B  ^ c  C )  =  ( 0  ^ c  C ) )
1110oveq2d 5876 . . . 4  |-  ( B  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( A  ^ c  C
)  x.  ( 0  ^ c  C ) ) )
129, 11eqeq12d 2299 . . 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 8863 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  B  e.  CC )
1615mul02d 9012 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  x.  B )  =  0 )
1716oveq1d 5875 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( 0  x.  B )  ^ c  C )  =  ( 0  ^ c  C
) )
1815, 5mulcxplem 20033 . . . . . . 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 20023 . . . . . . . . 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 8833 . . . . . . . . 9  |-  0  e.  CC
22 cxpcl 20023 . . . . . . . . 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 8858 . . . . . . 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 2317 . . . . . 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 2317 . . . . 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 5867 . . . . . . 7  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
2827oveq1d 5875 . . . . . 6  |-  ( A  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( 0  x.  B )  ^ c  C ) )
29 oveq1 5867 . . . . . . 7  |-  ( A  =  0  ->  ( A  ^ c  C )  =  ( 0  ^ c  C ) )
3029oveq1d 5875 . . . . . 6  |-  ( A  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( 0  ^ c  C
)  x.  ( B  ^ c  C ) ) )
3128, 30eqeq12d 2299 . . . . 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 8958 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  A )
3834, 37elrpd 10390 . . . . . . . . . 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 8958 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  B )
4339, 42elrpd 10390 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  RR+ )
4438, 43relogmuld 19978 . . . . . . . . 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 5876 . . . . . . . 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 19928 . . . . . . . . . 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 19928 . . . . . . . . . 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 8861 . . . . . . . 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 2317 . . . . . . 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 5531 . . . . . 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 8857 . . . . . . 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 8857 . . . . . . 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 12377 . . . . . . 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 2317 . . . . 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 8857 . . . . . 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 9422 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
63 cxpef 20014 . . . . . 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 20014 . . . . . . 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 20014 . . . . . . 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 5878 . . . . 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 2327 . . . 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 2525 . 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 2525 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 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739    + caddc 8742    x. cmul 8744    <_ cle 8870   expce 12345   logclog 19914    ^ c ccxp 19915
This theorem is referenced by:  cxprec  20035  divcxp  20036  mulcxpd  20077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-cxp 19917
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