MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isosctrlem3 Structured version   Unicode version

Theorem isosctrlem3 22177
Description: Lemma for isosctr 22178. 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 1007 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  e.  CC )
2 simp21 1016 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  0 )
3 simp1r 1008 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  e.  CC )
41, 3subcld 9715 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  -  B )  e.  CC )
5 simp23 1018 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  B )
61, 3, 5subne0d 9724 . . 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 22158 . . 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 1214 . 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 9731 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( A  -  B )  =  ( B  -  A ) )
1110oveq2d 6106 . 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 9398 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  e.  CC )
13 ax-1ne0 9347 . . . . . 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 10103 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  e.  CC )
1612, 15subcld 9715 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1  -  ( B  /  A ) )  e.  CC )
175necomd 2693 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  A )
183, 1, 2, 17divne1d 10114 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  1 )
1918necomd 2693 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  =/=  ( B  /  A
) )
2012, 15, 19subne0d 9724 . . . . 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 22159 . . . 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 10095 . . . . . 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 5692 . . . . 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 5692 . . . 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 12937 . . . . . . 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 985 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =  ( abs `  B
) )
2726eqcomd 2446 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  B )  =  ( abs `  A
) )
2827oveq1d 6105 . . . . . . 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 12918 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  RR )
3029recnd 9408 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  CC )
311, 2absne0d 12929 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =/=  0 )
3230, 31dividd 10101 . . . . . . 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 2477 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  ( B  /  A ) )  =  1 )
3419neneqd 2622 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -.  1  =  ( B  /  A ) )
35 isosctrlem2 22176 . . . . . 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 1213 . . . . 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 9702 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( B  /  A )  e.  CC )
38 simp22 1017 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  0 )
393, 1, 38, 2divne0d 10119 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  0 )
4015, 39negne0d 9713 . . . . . 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 22159 . . . . 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 2476 . . . 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 2477 . . 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 9399 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  1 )  =  A )
451, 12, 15subdid 9796 . . . . . 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 10105 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  ( B  /  A ) )  =  B )
4744, 46oveq12d 6108 . . . . . 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 2473 . . . . 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 6108 . . . 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 22157 . . . . 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 1229 . . . 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 2475 . . 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 9794 . . . . . 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 9600 . . . . . 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 2473 . . . . 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 6108 . . . 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 22157 . . . . 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 1229 . . . 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 2475 . . 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 2483 . 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 2481 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322   {csn 3874   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283    - cmin 9591   -ucneg 9592    / cdiv 9989   Imcim 12583   abscabs 12719   logclog 21965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967
This theorem is referenced by:  isosctr  22178
  Copyright terms: Public domain W3C validator