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Theorem cosknegpi 31851
Description: The cosine of an integer multiple of negative  pi is either  1 ore negative  1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
cosknegpi  |-  ( K  e.  ZZ  ->  ( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K ,  1 ,  -u 1 ) )

Proof of Theorem cosknegpi
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( ( K  e.  ZZ  /\  2  ||  K )  -> 
2  ||  K )
2 2z 10917 . . . . 5  |-  2  e.  ZZ
3 simpl 457 . . . . 5  |-  ( ( K  e.  ZZ  /\  2  ||  K )  ->  K  e.  ZZ )
4 divides 14000 . . . . 5  |-  ( ( 2  e.  ZZ  /\  K  e.  ZZ )  ->  ( 2  ||  K  <->  E. n  e.  ZZ  (
n  x.  2 )  =  K ) )
52, 3, 4sylancr 663 . . . 4  |-  ( ( K  e.  ZZ  /\  2  ||  K )  -> 
( 2  ||  K  <->  E. n  e.  ZZ  (
n  x.  2 )  =  K ) )
61, 5mpbid 210 . . 3  |-  ( ( K  e.  ZZ  /\  2  ||  K )  ->  E. n  e.  ZZ  ( n  x.  2
)  =  K )
7 zcn 10890 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  n  e.  CC )
8 negcl 9839 . . . . . . . . . . . . . . 15  |-  ( n  e.  CC  ->  -u n  e.  CC )
9 2cn 10627 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
10 picn 22978 . . . . . . . . . . . . . . . . 17  |-  pi  e.  CC
119, 10mulcli 9618 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  pi )  e.  CC
1211a1i 11 . . . . . . . . . . . . . . 15  |-  ( n  e.  CC  ->  (
2  x.  pi )  e.  CC )
138, 12mulcld 9633 . . . . . . . . . . . . . 14  |-  ( n  e.  CC  ->  ( -u n  x.  ( 2  x.  pi ) )  e.  CC )
1413addid2d 9798 . . . . . . . . . . . . 13  |-  ( n  e.  CC  ->  (
0  +  ( -u n  x.  ( 2  x.  pi ) ) )  =  ( -u n  x.  ( 2  x.  pi ) ) )
15 2cnd 10629 . . . . . . . . . . . . . . 15  |-  ( n  e.  CC  ->  2  e.  CC )
1610a1i 11 . . . . . . . . . . . . . . 15  |-  ( n  e.  CC  ->  pi  e.  CC )
178, 15, 16mulassd 9636 . . . . . . . . . . . . . 14  |-  ( n  e.  CC  ->  (
( -u n  x.  2 )  x.  pi )  =  ( -u n  x.  ( 2  x.  pi ) ) )
1817eqcomd 2465 . . . . . . . . . . . . 13  |-  ( n  e.  CC  ->  ( -u n  x.  ( 2  x.  pi ) )  =  ( ( -u n  x.  2 )  x.  pi ) )
19 id 22 . . . . . . . . . . . . . . 15  |-  ( n  e.  CC  ->  n  e.  CC )
2019, 15mulneg1d 10030 . . . . . . . . . . . . . 14  |-  ( n  e.  CC  ->  ( -u n  x.  2 )  =  -u ( n  x.  2 ) )
2120oveq1d 6311 . . . . . . . . . . . . 13  |-  ( n  e.  CC  ->  (
( -u n  x.  2 )  x.  pi )  =  ( -u (
n  x.  2 )  x.  pi ) )
2214, 18, 213eqtrd 2502 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  (
0  +  ( -u n  x.  ( 2  x.  pi ) ) )  =  ( -u ( n  x.  2
)  x.  pi ) )
237, 22syl 16 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
0  +  ( -u n  x.  ( 2  x.  pi ) ) )  =  ( -u ( n  x.  2
)  x.  pi ) )
2423adantr 465 . . . . . . . . . 10  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  K )  ->  ( 0  +  ( -u n  x.  ( 2  x.  pi ) ) )  =  ( -u ( n  x.  2 )  x.  pi ) )
2519, 15mulcld 9633 . . . . . . . . . . . . 13  |-  ( n  e.  CC  ->  (
n  x.  2 )  e.  CC )
26 mulneg12 10016 . . . . . . . . . . . . 13  |-  ( ( ( n  x.  2 )  e.  CC  /\  pi  e.  CC )  -> 
( -u ( n  x.  2 )  x.  pi )  =  ( (
n  x.  2 )  x.  -u pi ) )
2725, 16, 26syl2anc 661 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  ( -u ( n  x.  2 )  x.  pi )  =  ( ( n  x.  2 )  x.  -u pi ) )
287, 27syl 16 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  ( -u ( n  x.  2 )  x.  pi )  =  ( ( n  x.  2 )  x.  -u pi ) )
2928adantr 465 . . . . . . . . . 10  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  K )  ->  ( -u (
n  x.  2 )  x.  pi )  =  ( ( n  x.  2 )  x.  -u pi ) )
30 oveq1 6303 . . . . . . . . . . 11  |-  ( ( n  x.  2 )  =  K  ->  (
( n  x.  2 )  x.  -u pi )  =  ( K  x.  -u pi ) )
3130adantl 466 . . . . . . . . . 10  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  K )  ->  ( ( n  x.  2 )  x.  -u pi )  =  ( K  x.  -u pi ) )
3224, 29, 313eqtrrd 2503 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  K )  ->  ( K  x.  -u pi )  =  ( 0  +  ( -u n  x.  ( 2  x.  pi ) ) ) )
3332fveq2d 5876 . . . . . . . 8  |-  ( ( n  e.  ZZ  /\  ( n  x.  2
)  =  K )  ->  ( cos `  ( K  x.  -u pi ) )  =  ( cos `  ( 0  +  (
-u n  x.  (
2  x.  pi ) ) ) ) )
34333adant1 1014 . . . . . . 7  |-  ( ( 2  ||  K  /\  n  e.  ZZ  /\  (
n  x.  2 )  =  K )  -> 
( cos `  ( K  x.  -u pi ) )  =  ( cos `  ( 0  +  (
-u n  x.  (
2  x.  pi ) ) ) ) )
35 0cnd 9606 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  0  e.  CC )
36 znegcl 10920 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  -u n  e.  ZZ )
37 cosper 23001 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  -u n  e.  ZZ )  ->  ( cos `  (
0  +  ( -u n  x.  ( 2  x.  pi ) ) ) )  =  ( cos `  0 ) )
3835, 36, 37syl2anc 661 . . . . . . . 8  |-  ( n  e.  ZZ  ->  ( cos `  ( 0  +  ( -u n  x.  ( 2  x.  pi ) ) ) )  =  ( cos `  0
) )
39383ad2ant2 1018 . . . . . . 7  |-  ( ( 2  ||  K  /\  n  e.  ZZ  /\  (
n  x.  2 )  =  K )  -> 
( cos `  (
0  +  ( -u n  x.  ( 2  x.  pi ) ) ) )  =  ( cos `  0 ) )
40 iftrue 3950 . . . . . . . . 9  |-  ( 2 
||  K  ->  if ( 2  ||  K ,  1 ,  -u
1 )  =  1 )
41 cos0 13897 . . . . . . . . 9  |-  ( cos `  0 )  =  1
4240, 41syl6reqr 2517 . . . . . . . 8  |-  ( 2 
||  K  ->  ( cos `  0 )  =  if ( 2  ||  K ,  1 ,  -u 1 ) )
43423ad2ant1 1017 . . . . . . 7  |-  ( ( 2  ||  K  /\  n  e.  ZZ  /\  (
n  x.  2 )  =  K )  -> 
( cos `  0
)  =  if ( 2  ||  K , 
1 ,  -u 1
) )
4434, 39, 433eqtrd 2502 . . . . . 6  |-  ( ( 2  ||  K  /\  n  e.  ZZ  /\  (
n  x.  2 )  =  K )  -> 
( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K , 
1 ,  -u 1
) )
45443exp 1195 . . . . 5  |-  ( 2 
||  K  ->  (
n  e.  ZZ  ->  ( ( n  x.  2 )  =  K  -> 
( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K , 
1 ,  -u 1
) ) ) )
4645adantl 466 . . . 4  |-  ( ( K  e.  ZZ  /\  2  ||  K )  -> 
( n  e.  ZZ  ->  ( ( n  x.  2 )  =  K  ->  ( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K , 
1 ,  -u 1
) ) ) )
4746rexlimdv 2947 . . 3  |-  ( ( K  e.  ZZ  /\  2  ||  K )  -> 
( E. n  e.  ZZ  ( n  x.  2 )  =  K  ->  ( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K , 
1 ,  -u 1
) ) )
486, 47mpd 15 . 2  |-  ( ( K  e.  ZZ  /\  2  ||  K )  -> 
( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K , 
1 ,  -u 1
) )
49 odd2np1 14058 . . . 4  |-  ( K  e.  ZZ  ->  ( -.  2  ||  K  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  K ) )
5049biimpa 484 . . 3  |-  ( ( K  e.  ZZ  /\  -.  2  ||  K )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  K )
51 oveq1 6303 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  +  1 )  =  K  ->  (
( ( 2  x.  n )  +  1 )  x.  -u pi )  =  ( K  x.  -u pi ) )
5251eqcomd 2465 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  +  1 )  =  K  ->  ( K  x.  -u pi )  =  ( ( ( 2  x.  n )  +  1 )  x.  -u pi ) )
5352adantl 466 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( K  x.  -u pi )  =  ( ( ( 2  x.  n )  +  1 )  x.  -u pi ) )
5415, 19mulcld 9633 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  (
2  x.  n )  e.  CC )
55 1cnd 9629 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  1  e.  CC )
56 negpicn 22981 . . . . . . . . . . . . 13  |-  -u pi  e.  CC
5756a1i 11 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  -u pi  e.  CC )
5854, 55, 57adddird 9638 . . . . . . . . . . 11  |-  ( n  e.  CC  ->  (
( ( 2  x.  n )  +  1 )  x.  -u pi )  =  ( (
( 2  x.  n
)  x.  -u pi )  +  ( 1  x.  -u pi ) ) )
597, 58syl 16 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  (
( ( 2  x.  n )  +  1 )  x.  -u pi )  =  ( (
( 2  x.  n
)  x.  -u pi )  +  ( 1  x.  -u pi ) ) )
6059adantr 465 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( ( ( 2  x.  n )  +  1 )  x.  -u pi )  =  ( ( ( 2  x.  n )  x.  -u pi )  +  ( 1  x.  -u pi ) ) )
61 mulneg12 10016 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2  x.  n
)  e.  CC  /\  pi  e.  CC )  -> 
( -u ( 2  x.  n )  x.  pi )  =  ( (
2  x.  n )  x.  -u pi ) )
6254, 16, 61syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( n  e.  CC  ->  ( -u ( 2  x.  n
)  x.  pi )  =  ( ( 2  x.  n )  x.  -u pi ) )
6362eqcomd 2465 . . . . . . . . . . . . . 14  |-  ( n  e.  CC  ->  (
( 2  x.  n
)  x.  -u pi )  =  ( -u (
2  x.  n )  x.  pi ) )
6415, 19mulneg2d 10031 . . . . . . . . . . . . . . . 16  |-  ( n  e.  CC  ->  (
2  x.  -u n
)  =  -u (
2  x.  n ) )
6515, 8mulcomd 9634 . . . . . . . . . . . . . . . 16  |-  ( n  e.  CC  ->  (
2  x.  -u n
)  =  ( -u n  x.  2 ) )
6664, 65eqtr3d 2500 . . . . . . . . . . . . . . 15  |-  ( n  e.  CC  ->  -u (
2  x.  n )  =  ( -u n  x.  2 ) )
6766oveq1d 6311 . . . . . . . . . . . . . 14  |-  ( n  e.  CC  ->  ( -u ( 2  x.  n
)  x.  pi )  =  ( ( -u n  x.  2 )  x.  pi ) )
6863, 67, 173eqtrd 2502 . . . . . . . . . . . . 13  |-  ( n  e.  CC  ->  (
( 2  x.  n
)  x.  -u pi )  =  ( -u n  x.  ( 2  x.  pi ) ) )
6957mulid2d 9631 . . . . . . . . . . . . 13  |-  ( n  e.  CC  ->  (
1  x.  -u pi )  =  -u pi )
7068, 69oveq12d 6314 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  (
( ( 2  x.  n )  x.  -u pi )  +  ( 1  x.  -u pi ) )  =  ( ( -u n  x.  ( 2  x.  pi ) )  +  -u pi ) )
7113, 57addcomd 9799 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  (
( -u n  x.  (
2  x.  pi ) )  +  -u pi )  =  ( -u pi  +  ( -u n  x.  ( 2  x.  pi ) ) ) )
7270, 71eqtrd 2498 . . . . . . . . . . 11  |-  ( n  e.  CC  ->  (
( ( 2  x.  n )  x.  -u pi )  +  ( 1  x.  -u pi ) )  =  ( -u pi  +  ( -u n  x.  ( 2  x.  pi ) ) ) )
737, 72syl 16 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  (
( ( 2  x.  n )  x.  -u pi )  +  ( 1  x.  -u pi ) )  =  ( -u pi  +  ( -u n  x.  ( 2  x.  pi ) ) ) )
7473adantr 465 . . . . . . . . 9  |-  ( ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( ( ( 2  x.  n )  x.  -u pi )  +  ( 1  x.  -u pi ) )  =  (
-u pi  +  (
-u n  x.  (
2  x.  pi ) ) ) )
7553, 60, 743eqtrd 2502 . . . . . . . 8  |-  ( ( n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( K  x.  -u pi )  =  (
-u pi  +  (
-u n  x.  (
2  x.  pi ) ) ) )
76753adant1 1014 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  n  e.  ZZ  /\  (
( 2  x.  n
)  +  1 )  =  K )  -> 
( K  x.  -u pi )  =  ( -u pi  +  ( -u n  x.  ( 2  x.  pi ) ) ) )
7776fveq2d 5876 . . . . . 6  |-  ( ( K  e.  ZZ  /\  n  e.  ZZ  /\  (
( 2  x.  n
)  +  1 )  =  K )  -> 
( cos `  ( K  x.  -u pi ) )  =  ( cos `  ( -u pi  +  ( -u n  x.  (
2  x.  pi ) ) ) ) )
78773adant1r 1221 . . . . 5  |-  ( ( ( K  e.  ZZ  /\ 
-.  2  ||  K
)  /\  n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( cos `  ( K  x.  -u pi ) )  =  ( cos `  ( -u pi  +  ( -u n  x.  ( 2  x.  pi ) ) ) ) )
79 cosper 23001 . . . . . . 7  |-  ( (
-u pi  e.  CC  /\  -u n  e.  ZZ )  ->  ( cos `  ( -u pi  +  ( -u n  x.  ( 2  x.  pi ) ) ) )  =  ( cos `  -u pi ) )
8056, 36, 79sylancr 663 . . . . . 6  |-  ( n  e.  ZZ  ->  ( cos `  ( -u pi  +  ( -u n  x.  ( 2  x.  pi ) ) ) )  =  ( cos `  -u pi ) )
81803ad2ant2 1018 . . . . 5  |-  ( ( ( K  e.  ZZ  /\ 
-.  2  ||  K
)  /\  n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( cos `  ( -u pi  +  ( -u n  x.  (
2  x.  pi ) ) ) )  =  ( cos `  -u pi ) )
82 iffalse 3953 . . . . . . . 8  |-  ( -.  2  ||  K  ->  if ( 2  ||  K ,  1 ,  -u
1 )  =  -u
1 )
83 cosnegpi 31849 . . . . . . . 8  |-  ( cos `  -u pi )  = 
-u 1
8482, 83syl6reqr 2517 . . . . . . 7  |-  ( -.  2  ||  K  -> 
( cos `  -u pi )  =  if (
2  ||  K , 
1 ,  -u 1
) )
8584adantl 466 . . . . . 6  |-  ( ( K  e.  ZZ  /\  -.  2  ||  K )  ->  ( cos `  -u pi )  =  if (
2  ||  K , 
1 ,  -u 1
) )
86853ad2ant1 1017 . . . . 5  |-  ( ( ( K  e.  ZZ  /\ 
-.  2  ||  K
)  /\  n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( cos `  -u pi )  =  if ( 2  ||  K ,  1 ,  -u
1 ) )
8778, 81, 863eqtrd 2502 . . . 4  |-  ( ( ( K  e.  ZZ  /\ 
-.  2  ||  K
)  /\  n  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  =  K )  ->  ( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K ,  1 ,  -u
1 ) )
8887rexlimdv3a 2951 . . 3  |-  ( ( K  e.  ZZ  /\  -.  2  ||  K )  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  K  ->  ( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K , 
1 ,  -u 1
) ) )
8950, 88mpd 15 . 2  |-  ( ( K  e.  ZZ  /\  -.  2  ||  K )  ->  ( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K , 
1 ,  -u 1
) )
9048, 89pm2.61dan 791 1  |-  ( K  e.  ZZ  ->  ( cos `  ( K  x.  -u pi ) )  =  if ( 2  ||  K ,  1 ,  -u 1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E.wrex 2808   ifcif 3944   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   -ucneg 9825   2c2 10606   ZZcz 10885   cosccos 13812   picpi 13814    || cdvds 13998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-pi 13820  df-dvds 13999  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397
This theorem is referenced by:  sqwvfourb  32194
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