MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ang180lem4 Unicode version

Theorem ang180lem4 20110
Description: Lemma for ang180 20112. Reduce the statement to one variable. (Contributed by Mario Carneiro, 23-Sep-2014.)
Hypothesis
Ref Expression
ang.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
Assertion
Ref Expression
ang180lem4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  e.  { -u pi ,  pi } )
Distinct variable group:    x, y, A
Allowed substitution hints:    F( x, y)

Proof of Theorem ang180lem4
StepHypRef Expression
1 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
21a1i 10 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  e.  CC )
3 simp1 955 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
42, 3subcld 9157 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
5 simp3 957 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
65necomd 2529 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
7 subeq0 9073 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
82, 3, 7syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
98necon3bid 2481 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =/=  0  <->  1  =/=  A ) )
106, 9mpbird 223 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
11 ax-1ne0 8806 . . . . . . . 8  |-  1  =/=  0
1211a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  0 )
13 ang.1 . . . . . . . 8  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
1413angval 20099 . . . . . . 7  |-  ( ( ( ( 1  -  A )  e.  CC  /\  ( 1  -  A
)  =/=  0 )  /\  ( 1  e.  CC  /\  1  =/=  0 ) )  -> 
( ( 1  -  A ) F 1 )  =  ( Im
`  ( log `  (
1  /  ( 1  -  A ) ) ) ) )
154, 10, 2, 12, 14syl22anc 1183 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
) F 1 )  =  ( Im `  ( log `  ( 1  /  ( 1  -  A ) ) ) ) )
16 simp2 956 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
173, 2subcld 9157 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
18 subeq0 9073 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
193, 2, 18syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
2019necon3bid 2481 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =/=  0  <->  A  =/=  1 ) )
215, 20mpbird 223 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
2213angval 20099 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( ( A  -  1 )  e.  CC  /\  ( A  -  1 )  =/=  0 ) )  -> 
( A F ( A  -  1 ) )  =  ( Im
`  ( log `  (
( A  -  1 )  /  A ) ) ) )
233, 16, 17, 21, 22syl22anc 1183 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A F ( A  - 
1 ) )  =  ( Im `  ( log `  ( ( A  -  1 )  /  A ) ) ) )
2415, 23oveq12d 5876 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( 1  -  A ) F 1 )  +  ( A F ( A  - 
1 ) ) )  =  ( ( Im
`  ( log `  (
1  /  ( 1  -  A ) ) ) )  +  ( Im `  ( log `  ( ( A  - 
1 )  /  A
) ) ) ) )
252, 4, 10divcld 9536 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
264, 10recne0d 9530 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
27 logcl 19926 . . . . . . 7  |-  ( ( ( 1  /  (
1  -  A ) )  e.  CC  /\  ( 1  /  (
1  -  A ) )  =/=  0 )  ->  ( log `  (
1  /  ( 1  -  A ) ) )  e.  CC )
2825, 26, 27syl2anc 642 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
2917, 3, 16divcld 9536 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
3017, 3, 21, 16divne0d 9552 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
31 logcl 19926 . . . . . . 7  |-  ( ( ( ( A  - 
1 )  /  A
)  e.  CC  /\  ( ( A  - 
1 )  /  A
)  =/=  0 )  ->  ( log `  (
( A  -  1 )  /  A ) )  e.  CC )
3229, 30, 31syl2anc 642 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
3328, 32imaddd 11700 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( ( log `  ( 1  / 
( 1  -  A
) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) ) )  =  ( ( Im `  ( log `  ( 1  /  ( 1  -  A ) ) ) )  +  ( Im
`  ( log `  (
( A  -  1 )  /  A ) ) ) ) )
3424, 33eqtr4d 2318 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( 1  -  A ) F 1 )  +  ( A F ( A  - 
1 ) ) )  =  ( Im `  ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) ) )
3513angval 20099 . . . . . 6  |-  ( ( ( 1  e.  CC  /\  1  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( 1 F A )  =  ( Im
`  ( log `  ( A  /  1 ) ) ) )
362, 12, 3, 16, 35syl22anc 1183 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  ( A  /  1 ) ) ) )
373div1d 9528 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  /  1 )  =  A )
3837fveq2d 5529 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( A  / 
1 ) )  =  ( log `  A
) )
3938fveq2d 5529 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( log `  ( A  /  1
) ) )  =  ( Im `  ( log `  A ) ) )
4036, 39eqtrd 2315 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  A ) ) )
4134, 40oveq12d 5876 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  =  ( ( Im
`  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  +  ( Im `  ( log `  A ) ) ) )
4228, 32addcld 8854 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
43 logcl 19926 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
443, 16, 43syl2anc 642 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
4542, 44imaddd 11700 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  =  ( ( Im `  ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) )  +  ( Im `  ( log `  A ) ) ) )
4641, 45eqtr4d 2318 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  =  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) ) )
47 eqid 2283 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )
48 eqid 2283 . . . 4  |-  ( ( ( ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) )  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  =  ( ( ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )
4913, 47, 48ang180lem3 20109 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) } )
50 fveq2 5525 . . . . . 6  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  -u ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  ( Im `  -u ( _i  x.  pi ) ) )
51 ax-icn 8796 . . . . . . . . 9  |-  _i  e.  CC
52 pire 19832 . . . . . . . . . 10  |-  pi  e.  RR
5352recni 8849 . . . . . . . . 9  |-  pi  e.  CC
5451, 53mulcli 8842 . . . . . . . 8  |-  ( _i  x.  pi )  e.  CC
5554imnegi 11666 . . . . . . 7  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u ( Im `  ( _i  x.  pi ) )
5654addid2i 9000 . . . . . . . . . 10  |-  ( 0  +  ( _i  x.  pi ) )  =  ( _i  x.  pi )
5756fveq2i 5528 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  ( Im `  (
_i  x.  pi )
)
58 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
5958, 52crimi 11678 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  pi
6057, 59eqtr3i 2305 . . . . . . . 8  |-  ( Im
`  ( _i  x.  pi ) )  =  pi
6160negeqi 9045 . . . . . . 7  |-  -u (
Im `  ( _i  x.  pi ) )  = 
-u pi
6255, 61eqtri 2303 . . . . . 6  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u pi
6350, 62syl6eq 2331 . . . . 5  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  -u ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u pi )
64 fveq2 5525 . . . . . 6  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  ( Im `  ( _i  x.  pi ) ) )
6564, 60syl6eq 2331 . . . . 5  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  pi )
6663, 65orim12i 502 . . . 4  |-  ( ( ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) )  = 
-u ( _i  x.  pi )  \/  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi ) )  ->  ( (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  = 
-u pi  \/  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  =  pi ) )
67 ovex 5883 . . . . 5  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
6867elpr 3658 . . . 4  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) }  <->  ( (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  -u ( _i  x.  pi )  \/  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi ) ) )
69 fvex 5539 . . . . 5  |-  ( Im
`  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  e.  _V
7069elpr 3658 . . . 4  |-  ( ( Im `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  e. 
{ -u pi ,  pi } 
<->  ( ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u pi  \/  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  =  pi ) )
7166, 68, 703imtr4i 257 . . 3  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) }  ->  ( Im `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  e. 
{ -u pi ,  pi } )
7249, 71syl 15 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  e. 
{ -u pi ,  pi } )
7346, 72eqeltrd 2357 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  e.  { -u pi ,  pi } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   Imcim 11583   picpi 12348   logclog 19912
This theorem is referenced by:  ang180lem5  20111
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
  Copyright terms: Public domain W3C validator