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Theorem isosctrlem3 20120
Description: Lemma for isosctr 20121. 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 979 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  e.  CC )
2 simp21 988 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  0 )
3 simp1r 980 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  e.  CC )
41, 3subcld 9157 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  -  B )  e.  CC )
5 simp23 990 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  A  =/=  B )
6 subeq0 9073 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  0  <-> 
A  =  B ) )
71, 3, 6syl2anc 642 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  -  B
)  =  0  <->  A  =  B ) )
87necon3bid 2481 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( A  -  B
)  =/=  0  <->  A  =/=  B ) )
95, 8mpbird 223 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  -  B )  =/=  0 )
10 isosctrlem3.1 . . . 4  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
1110angneg 20101 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( ( A  -  B )  e.  CC  /\  ( A  -  B )  =/=  0 ) )  -> 
( -u A F -u ( A  -  B
) )  =  ( A F ( A  -  B ) ) )
121, 2, 4, 9, 11syl22anc 1183 . 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 ) ) )
131, 3negsubdi2d 9173 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( A  -  B )  =  ( B  -  A ) )
1413oveq2d 5874 . 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 ) ) )
15 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
1615a1i 10 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  e.  CC )
17 ax-1ne0 8806 . . . . . 6  |-  1  =/=  0
1817a1i 10 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  =/=  0 )
193, 1, 2divcld 9536 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  e.  CC )
2016, 19subcld 9157 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1  -  ( B  /  A ) )  e.  CC )
215necomd 2529 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  A )
221mulid1d 8852 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  1 )  =  A )
2321, 22neeqtrrd 2470 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  ( A  x.  1 ) )
243, 16, 1, 2divmul2d 9569 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( B  /  A
)  =  1  <->  B  =  ( A  x.  1 ) ) )
2524necon3bid 2481 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( B  /  A
)  =/=  1  <->  B  =/=  ( A  x.  1 ) ) )
2623, 25mpbird 223 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  1 )
2726necomd 2529 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  1  =/=  ( B  /  A
) )
28 subeq0 9073 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( B  /  A
)  e.  CC )  ->  ( ( 1  -  ( B  /  A ) )  =  0  <->  1  =  ( B  /  A ) ) )
2916, 19, 28syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( 1  -  ( B  /  A ) )  =  0  <->  1  =  ( B  /  A
) ) )
3029necon3bid 2481 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( 1  -  ( B  /  A ) )  =/=  0  <->  1  =/=  ( B  /  A
) ) )
3127, 30mpbird 223 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
1  -  ( B  /  A ) )  =/=  0 )
3210angval 20099 . . . . 5  |-  ( ( ( 1  e.  CC  /\  1  =/=  0 )  /\  ( ( 1  -  ( B  /  A ) )  e.  CC  /\  ( 1  -  ( B  /  A ) )  =/=  0 ) )  -> 
( 1 F ( 1  -  ( B  /  A ) ) )  =  ( Im
`  ( log `  (
( 1  -  ( B  /  A ) )  /  1 ) ) ) )
3316, 18, 20, 31, 32syl22anc 1183 . . . 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 ) ) ) )
3420div1d 9528 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( 1  -  ( B  /  A ) )  /  1 )  =  ( 1  -  ( B  /  A ) ) )
3534fveq2d 5529 . . . . 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 ) ) ) )
3635fveq2d 5529 . . . 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
) ) ) ) )
373, 1, 2absdivd 11937 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  ( B  /  A ) )  =  ( ( abs `  B
)  /  ( abs `  A ) ) )
38 simp3 957 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =  ( abs `  B
) )
3938eqcomd 2288 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  B )  =  ( abs `  A
) )
4039oveq1d 5873 . . . . . . 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 ) ) )
411abscld 11918 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  RR )
4241recnd 8861 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  e.  CC )
431, 2absne0d 11929 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  A )  =/=  0 )
4442, 43dividd 9534 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  (
( abs `  A
)  /  ( abs `  A ) )  =  1 )
4537, 40, 443eqtrd 2319 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( abs `  ( B  /  A ) )  =  1 )
4627neneqd 2462 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -.  1  =  ( B  /  A ) )
47 isosctrlem2 20119 . . . . . 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 ) ) ) ) ) )
4819, 45, 46, 47syl3anc 1182 . . . . 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 ) ) ) ) ) )
4919negcld 9144 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( B  /  A )  e.  CC )
50 simp22 989 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  B  =/=  0 )
513, 1, 50, 2divne0d 9552 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( B  /  A )  =/=  0 )
5219, 51negne0d 9155 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( B  /  A )  =/=  0 )
5310angval 20099 . . . . . 6  |-  ( ( ( ( 1  -  ( B  /  A
) )  e.  CC  /\  ( 1  -  ( B  /  A ) )  =/=  0 )  /\  ( -u ( B  /  A )  e.  CC  /\  -u ( B  /  A
)  =/=  0 ) )  ->  ( (
1  -  ( B  /  A ) ) F -u ( B  /  A ) )  =  ( Im `  ( log `  ( -u ( B  /  A
)  /  ( 1  -  ( B  /  A ) ) ) ) ) )
5420, 31, 49, 52, 53syl22anc 1183 . . . . 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 ) ) ) ) ) )
5548, 54eqtr4d 2318 . . . 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 ) ) )
5633, 36, 553eqtrd 2319 . . 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 ) ) )
571, 16, 19subdid 9235 . . . . . 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 ) ) ) )
583, 1, 2divcan2d 9538 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  ( B  /  A ) )  =  B )
5922, 58oveq12d 5876 . . . . . 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 ) )
6057, 59eqtrd 2315 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  ( 1  -  ( B  /  A ) ) )  =  ( A  -  B ) )
6122, 60oveq12d 5876 . . . 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
) ) )
6210angcan 20100 . . . . 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
) ) ) )
6316, 18, 20, 31, 1, 2, 62syl222anc 1198 . . . 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 ) ) ) )
6461, 63eqtr3d 2317 . . 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 ) ) ) )
651, 19mulneg2d 9233 . . . . . 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
) ) )
6658negeqd 9046 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  -u ( A  x.  ( B  /  A ) )  = 
-u B )
6765, 66eqtrd 2315 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  =/=  0  /\  B  =/=  0  /\  A  =/= 
B )  /\  ( abs `  A )  =  ( abs `  B
) )  ->  ( A  x.  -u ( B  /  A ) )  =  -u B )
6860, 67oveq12d 5876 . . . 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 ) )
6910angcan 20100 . . . . 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 ) ) )
7020, 31, 49, 52, 1, 2, 69syl222anc 1198 . . . 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 ) ) )
7168, 70eqtr3d 2317 . . 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 ) ) )
7256, 64, 713eqtr4d 2325 . 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 ) )
7312, 14, 723eqtr3d 2323 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   Imcim 11583   abscabs 11719   logclog 19912
This theorem is referenced by:  isosctr  20121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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