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Theorem mulcxp 19995
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 8829 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  A  e.  CC )
32mul01d 8979 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  x.  0 )  =  0 )
43oveq1d 5807 . . . 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 19994 . . . 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 2290 . . 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 5800 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
98oveq1d 5807 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( A  x.  0 )  ^ c  C ) )
10 oveq1 5799 . . . . 5  |-  ( B  =  0  ->  ( B  ^ c  C )  =  ( 0  ^ c  C ) )
1110oveq2d 5808 . . . 4  |-  ( B  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( A  ^ c  C
)  x.  ( 0  ^ c  C ) ) )
129, 11eqeq12d 2272 . . 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 8829 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  B  e.  CC )
1615mul02d 8978 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  x.  B )  =  0 )
1716oveq1d 5807 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( 0  x.  B )  ^ c  C )  =  ( 0  ^ c  C
) )
1815, 5mulcxplem 19994 . . . . . . 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 19984 . . . . . . . . 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 8799 . . . . . . . . 9  |-  0  e.  CC
22 cxpcl 19984 . . . . . . . . 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 8824 . . . . . . 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 2290 . . . . . 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 2290 . . . . 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 5799 . . . . . . 7  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
2827oveq1d 5807 . . . . . 6  |-  ( A  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( 0  x.  B )  ^ c  C ) )
29 oveq1 5799 . . . . . . 7  |-  ( A  =  0  ->  ( A  ^ c  C )  =  ( 0  ^ c  C ) )
3029oveq1d 5807 . . . . . 6  |-  ( A  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( 0  ^ c  C
)  x.  ( B  ^ c  C ) ) )
3128, 30eqeq12d 2272 . . . . 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 8924 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  A )
3834, 37elrpd 10356 . . . . . . . . . 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 8924 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  B )
4339, 42elrpd 10356 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  RR+ )
4438, 43relogmuld 19939 . . . . . . . . 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 5808 . . . . . . . 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 19889 . . . . . . . . . 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 19889 . . . . . . . . . 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 8827 . . . . . . . 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 2290 . . . . . . 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 5462 . . . . . 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 8823 . . . . . . 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 8823 . . . . . . 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 12338 . . . . . . 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 2290 . . . . 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 8823 . . . . . 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 9388 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
63 cxpef 19975 . . . . . 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 19975 . . . . . . 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 19975 . . . . . . 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 5810 . . . . 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 2300 . . . 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 2498 . 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 2498 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 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705    + caddc 8708    x. cmul 8710    <_ cle 8836   expce 12306   logclog 19875    ^ c ccxp 19876
This theorem is referenced by:  cxprec  19996  divcxp  19997  mulcxpd  20038
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-ioo 10627  df-ioc 10628  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-fac 11256  df-bc 11283  df-hash 11305  df-shft 11528  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-limsup 11911  df-clim 11928  df-rlim 11929  df-sum 12125  df-ef 12312  df-sin 12314  df-cos 12315  df-pi 12317  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-rest 13290  df-topn 13291  df-topgen 13307  df-pt 13308  df-prds 13311  df-xrs 13366  df-0g 13367  df-gsum 13368  df-qtop 13373  df-imas 13374  df-xps 13376  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-mulg 14455  df-cntz 14756  df-cmn 15054  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cld 16719  df-ntr 16720  df-cls 16721  df-nei 16798  df-lp 16831  df-perf 16832  df-cn 16920  df-cnp 16921  df-haus 17006  df-tx 17220  df-hmeo 17409  df-fbas 17483  df-fg 17484  df-fil 17504  df-fm 17596  df-flim 17597  df-flf 17598  df-xms 17848  df-ms 17849  df-tms 17850  df-cncf 18345  df-limc 19179  df-dv 19180  df-log 19877  df-cxp 19878
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