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Theorem mulcxp 20143
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 8951 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  A  e.  CC )
32mul01d 9101 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( A  x.  0 )  =  0 )
43oveq1d 5960 . . . 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 20142 . . . 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 2390 . . 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 5953 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
98oveq1d 5960 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( A  x.  0 )  ^ c  C ) )
10 oveq1 5952 . . . . 5  |-  ( B  =  0  ->  ( B  ^ c  C )  =  ( 0  ^ c  C ) )
1110oveq2d 5961 . . . 4  |-  ( B  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( A  ^ c  C
)  x.  ( 0  ^ c  C ) ) )
129, 11eqeq12d 2372 . . 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 8951 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  B  e.  CC )
1615mul02d 9100 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( 0  x.  B )  =  0 )
1716oveq1d 5960 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  ->  ( ( 0  x.  B )  ^ c  C )  =  ( 0  ^ c  C
) )
1815, 5mulcxplem 20142 . . . . . . 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 20132 . . . . . . . . 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 8921 . . . . . . . . 9  |-  0  e.  CC
22 cxpcl 20132 . . . . . . . . 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 8946 . . . . . . 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 2390 . . . . . 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 2390 . . . . 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 5952 . . . . . . 7  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
2827oveq1d 5960 . . . . . 6  |-  ( A  =  0  ->  (
( A  x.  B
)  ^ c  C
)  =  ( ( 0  x.  B )  ^ c  C ) )
29 oveq1 5952 . . . . . . 7  |-  ( A  =  0  ->  ( A  ^ c  C )  =  ( 0  ^ c  C ) )
3029oveq1d 5960 . . . . . 6  |-  ( A  =  0  ->  (
( A  ^ c  C )  x.  ( B  ^ c  C ) )  =  ( ( 0  ^ c  C
)  x.  ( B  ^ c  C ) ) )
3128, 30eqeq12d 2372 . . . . 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 9046 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  A )
3834, 37elrpd 10480 . . . . . . . . . 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 9046 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
0  <  B )
4339, 42elrpd 10480 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  ->  B  e.  RR+ )
4438, 43relogmuld 20087 . . . . . . . . 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 5961 . . . . . . . 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 20033 . . . . . . . . . 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 20033 . . . . . . . . . 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 8949 . . . . . . . 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 2390 . . . . . . 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 5612 . . . . . 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 8945 . . . . . . 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 8945 . . . . . . 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 12472 . . . . . . 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 2390 . . . . 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 8945 . . . . . 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 9510 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
63 cxpef 20123 . . . . . 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 20123 . . . . . . 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 20123 . . . . . . 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 5963 . . . . 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 2400 . . . 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 2598 . 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 2598 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 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827    + caddc 8830    x. cmul 8832    <_ cle 8958   expce 12440   logclog 20019    ^ c ccxp 20020
This theorem is referenced by:  cxprec  20144  divcxp  20145  mulcxpd  20186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-ef 12446  df-sin 12448  df-cos 12449  df-pi 12451  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321  df-log 20021  df-cxp 20022
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