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Theorem isosctrlem3 22997
Description: Lemma for isosctr 22998. Corresponds to the case where one vertex is at 0. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
Hypothesis
Ref Expression
isosctrlem3.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
Assertion
Ref Expression
isosctrlem3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( -u A F ( B  -  A ) )  =  ( ( A  -  B ) F
-u B ) )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    F( x, y)

Proof of Theorem isosctrlem3
StepHypRef Expression
1 simp1l 1020 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  e.  CC )
2 simp21 1029 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  0 )
3 simp1r 1021 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  e.  CC )
41, 3subcld 9940 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  -  B )  e.  CC )
5 simp23 1031 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  B )
61, 3, 5subne0d 9949 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  -  B )  =/=  0 )
7 isosctrlem3.1 . . . 4  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
87angneg 22978 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( ( A  -  B )  e.  CC  /\  ( A  -  B )  =/=  0 ) )  -> 
( -u A F -u ( A  -  B
) )  =  ( A F ( A  -  B ) ) )
91, 2, 4, 6, 8syl22anc 1229 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( -u A F -u ( A  -  B )
)  =  ( A F ( A  -  B ) ) )
101, 3negsubdi2d 9956 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( A  -  B )  =  ( B  -  A ) )
1110oveq2d 6310 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( -u A F -u ( A  -  B )
)  =  ( -u A F ( B  -  A ) ) )
12 1cnd 9622 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  e.  CC )
13 ax-1ne0 9571 . . . . . 6  |-  1  =/=  0
1413a1i 11 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  =/=  0 )
153, 1, 2divcld 10330 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  e.  CC )
1612, 15subcld 9940 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1  -  ( B  /  A ) )  e.  CC )
175necomd 2738 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  A )
183, 1, 2, 17divne1d 10341 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  1 )
1918necomd 2738 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  =/=  ( B  /  A
) )
2012, 15, 19subne0d 9949 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1  -  ( B  /  A ) )  =/=  0 )
217, 12, 14, 16, 20angvald 22979 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1 F ( 1  -  ( B  /  A ) ) )  =  ( Im `  ( log `  ( ( 1  -  ( B  /  A ) )  /  1 ) ) ) )
2216div1d 10322 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( 1  -  ( B  /  A ) )  /  1 )  =  ( 1  -  ( B  /  A ) ) )
2322fveq2d 5875 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( log `  ( ( 1  -  ( B  /  A ) )  / 
1 ) )  =  ( log `  (
1  -  ( B  /  A ) ) ) )
2423fveq2d 5875 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
Im `  ( log `  ( ( 1  -  ( B  /  A
) )  /  1
) ) )  =  ( Im `  ( log `  ( 1  -  ( B  /  A
) ) ) ) )
253, 1, 2absdivd 13261 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  ( B  /  A ) )  =  ( ( abs `  B
)  /  ( abs `  A ) ) )
26 simp3 998 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =  ( abs `  B
) )
2726eqcomd 2475 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  B )  =  ( abs `  A
) )
2827oveq1d 6309 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( abs `  B
)  /  ( abs `  A ) )  =  ( ( abs `  A
)  /  ( abs `  A ) ) )
291abscld 13242 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  RR )
3029recnd 9632 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  CC )
311, 2absne0d 13253 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =/=  0 )
3230, 31dividd 10328 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( abs `  A
)  /  ( abs `  A ) )  =  1 )
3325, 28, 323eqtrd 2512 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  ( B  /  A ) )  =  1 )
3419neneqd 2669 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -.  1  =  ( B  /  A ) )
35 isosctrlem2 22996 . . . . . 6  |-  ( ( ( B  /  A
)  e.  CC  /\  ( abs `  ( B  /  A ) )  =  1  /\  -.  1  =  ( B  /  A ) )  -> 
( Im `  ( log `  ( 1  -  ( B  /  A
) ) ) )  =  ( Im `  ( log `  ( -u ( B  /  A
)  /  ( 1  -  ( B  /  A ) ) ) ) ) )
3615, 33, 34, 35syl3anc 1228 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
Im `  ( log `  ( 1  -  ( B  /  A ) ) ) )  =  ( Im `  ( log `  ( -u ( B  /  A )  / 
( 1  -  ( B  /  A ) ) ) ) ) )
3715negcld 9927 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( B  /  A )  e.  CC )
38 simp22 1030 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  0 )
393, 1, 38, 2divne0d 10346 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  0 )
4015, 39negne0d 9938 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( B  /  A )  =/=  0 )
417, 16, 20, 37, 40angvald 22979 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( 1  -  ( B  /  A ) ) F -u ( B  /  A ) )  =  ( Im `  ( log `  ( -u ( B  /  A
)  /  ( 1  -  ( B  /  A ) ) ) ) ) )
4236, 41eqtr4d 2511 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
Im `  ( log `  ( 1  -  ( B  /  A ) ) ) )  =  ( ( 1  -  ( B  /  A ) ) F -u ( B  /  A ) ) )
4321, 24, 423eqtrd 2512 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1 F ( 1  -  ( B  /  A ) ) )  =  ( ( 1  -  ( B  /  A ) ) F
-u ( B  /  A ) ) )
441mulid1d 9623 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  1 )  =  A )
451, 12, 15subdid 10022 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  ( 1  -  ( B  /  A ) ) )  =  ( ( A  x.  1 )  -  ( A  x.  ( B  /  A ) ) ) )
463, 1, 2divcan2d 10332 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  ( B  /  A ) )  =  B )
4744, 46oveq12d 6312 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  x.  1 )  -  ( A  x.  ( B  /  A ) ) )  =  ( A  -  B ) )
4845, 47eqtrd 2508 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  ( 1  -  ( B  /  A ) ) )  =  ( A  -  B ) )
4944, 48oveq12d 6312 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  x.  1 ) F ( A  x.  ( 1  -  ( B  /  A
) ) ) )  =  ( A F ( A  -  B
) ) )
507angcan 22977 . . . . 5  |-  ( ( ( 1  e.  CC  /\  1  =/=  0 )  /\  ( ( 1  -  ( B  /  A ) )  e.  CC  /\  ( 1  -  ( B  /  A ) )  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  1 ) F ( A  x.  ( 1  -  ( B  /  A ) ) ) )  =  ( 1 F ( 1  -  ( B  /  A
) ) ) )
5112, 14, 16, 20, 1, 2, 50syl222anc 1244 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  x.  1 ) F ( A  x.  ( 1  -  ( B  /  A
) ) ) )  =  ( 1 F ( 1  -  ( B  /  A ) ) ) )
5249, 51eqtr3d 2510 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A F ( A  -  B ) )  =  ( 1 F ( 1  -  ( B  /  A ) ) ) )
531, 15mulneg2d 10020 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  -u ( B  /  A ) )  =  -u ( A  x.  ( B  /  A
) ) )
5446negeqd 9824 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( A  x.  ( B  /  A ) )  = 
-u B )
5553, 54eqtrd 2508 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  -u ( B  /  A ) )  =  -u B )
5648, 55oveq12d 6312 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  x.  (
1  -  ( B  /  A ) ) ) F ( A  x.  -u ( B  /  A ) ) )  =  ( ( A  -  B ) F
-u B ) )
577angcan 22977 . . . . 5  |-  ( ( ( ( 1  -  ( B  /  A
) )  e.  CC  /\  ( 1  -  ( B  /  A ) )  =/=  0 )  /\  ( -u ( B  /  A )  e.  CC  /\  -u ( B  /  A
)  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  ( 1  -  ( B  /  A ) ) ) F ( A  x.  -u ( B  /  A ) ) )  =  ( ( 1  -  ( B  /  A ) ) F
-u ( B  /  A ) ) )
5816, 20, 37, 40, 1, 2, 57syl222anc 1244 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  x.  (
1  -  ( B  /  A ) ) ) F ( A  x.  -u ( B  /  A ) ) )  =  ( ( 1  -  ( B  /  A ) ) F
-u ( B  /  A ) ) )
5956, 58eqtr3d 2510 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  -  B
) F -u B
)  =  ( ( 1  -  ( B  /  A ) ) F -u ( B  /  A ) ) )
6043, 52, 593eqtr4d 2518 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A F ( A  -  B ) )  =  ( ( A  -  B ) F -u B ) )
619, 11, 603eqtr3d 2516 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( -u A F ( B  -  A ) )  =  ( ( A  -  B ) F
-u B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478   {csn 4032   ` cfv 5593  (class class class)co 6294    |-> cmpt2 6296   CCcc 9500   0cc0 9502   1c1 9503    x. cmul 9507    - cmin 9815   -ucneg 9816    / cdiv 10216   Imcim 12906   abscabs 13042   logclog 22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-addf 9581  ax-mulf 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-fi 7881  df-sup 7911  df-oi 7945  df-card 8330  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-ioc 11544  df-ico 11545  df-icc 11546  df-fz 11683  df-fzo 11803  df-fl 11907  df-mod 11975  df-seq 12086  df-exp 12145  df-fac 12332  df-bc 12359  df-hash 12384  df-shft 12875  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-limsup 13269  df-clim 13286  df-rlim 13287  df-sum 13484  df-ef 13677  df-sin 13679  df-cos 13680  df-pi 13682  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-starv 14582  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-unif 14590  df-hom 14591  df-cco 14592  df-rest 14690  df-topn 14691  df-0g 14709  df-gsum 14710  df-topgen 14711  df-pt 14712  df-prds 14715  df-xrs 14769  df-qtop 14774  df-imas 14775  df-xps 14777  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-submnd 15820  df-mulg 15909  df-cntz 16204  df-cmn 16650  df-psmet 18258  df-xmet 18259  df-met 18260  df-bl 18261  df-mopn 18262  df-fbas 18263  df-fg 18264  df-cnfld 18268  df-top 19245  df-bases 19247  df-topon 19248  df-topsp 19249  df-cld 19365  df-ntr 19366  df-cls 19367  df-nei 19444  df-lp 19482  df-perf 19483  df-cn 19573  df-cnp 19574  df-haus 19661  df-tx 19908  df-hmeo 20101  df-fil 20192  df-fm 20284  df-flim 20285  df-flf 20286  df-xms 20668  df-ms 20669  df-tms 20670  df-cncf 21227  df-limc 22115  df-dv 22116  df-log 22787
This theorem is referenced by:  isosctr  22998
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