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Theorem isosctrlem3 22218
Description: Lemma for isosctr 22219. 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 1012 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  e.  CC )
2 simp21 1021 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  0 )
3 simp1r 1013 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  e.  CC )
41, 3subcld 9719 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  -  B )  e.  CC )
5 simp23 1023 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  B )
61, 3, 5subne0d 9728 . . 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 22199 . . 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 1219 . 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 9735 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( A  -  B )  =  ( B  -  A ) )
1110oveq2d 6107 . 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 9402 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  e.  CC )
13 ax-1ne0 9351 . . . . . 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 10107 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  e.  CC )
1612, 15subcld 9719 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1  -  ( B  /  A ) )  e.  CC )
175necomd 2695 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  A )
183, 1, 2, 17divne1d 10118 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  1 )
1918necomd 2695 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  =/=  ( B  /  A
) )
2012, 15, 19subne0d 9728 . . . . 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 22200 . . . 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 10099 . . . . . 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 5695 . . . . 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 5695 . . . 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 12941 . . . . . . 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 990 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =  ( abs `  B
) )
2726eqcomd 2448 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  B )  =  ( abs `  A
) )
2827oveq1d 6106 . . . . . . 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 12922 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  RR )
3029recnd 9412 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  CC )
311, 2absne0d 12933 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =/=  0 )
3230, 31dividd 10105 . . . . . . 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 2479 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  ( B  /  A ) )  =  1 )
3419neneqd 2624 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -.  1  =  ( B  /  A ) )
35 isosctrlem2 22217 . . . . . 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 1218 . . . . 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 9706 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( B  /  A )  e.  CC )
38 simp22 1022 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  0 )
393, 1, 38, 2divne0d 10123 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  0 )
4015, 39negne0d 9717 . . . . . 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 22200 . . . . 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 2478 . . . 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 2479 . . 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 9403 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  1 )  =  A )
451, 12, 15subdid 9800 . . . . . 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 10109 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  ( B  /  A ) )  =  B )
4744, 46oveq12d 6109 . . . . . 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 2475 . . . . 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 6109 . . . 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 22198 . . . . 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 1234 . . . 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 2477 . . 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 9798 . . . . . 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 9604 . . . . . 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 2475 . . . . 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 6109 . . . 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 22198 . . . . 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 1234 . . . 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 2477 . . 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 2485 . 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 2483 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606    \ cdif 3325   {csn 3877   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   CCcc 9280   0cc0 9282   1c1 9283    x. cmul 9287    - cmin 9595   -ucneg 9596    / cdiv 9993   Imcim 12587   abscabs 12723   logclog 22006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008
This theorem is referenced by:  isosctr  22219
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