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

Theorem cnmsgnsubg 18113
Description: The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
Hypothesis
Ref Expression
cnmsgnsubg.m  |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
Assertion
Ref Expression
cnmsgnsubg  |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )

Proof of Theorem cnmsgnsubg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmsgnsubg.m . 2  |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
2 elpri 3992 . . 3  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( x  =  1  \/  x  =  -u 1
) )
3 id 22 . . . . 5  |-  ( x  =  1  ->  x  =  1 )
4 ax-1cn 9438 . . . . 5  |-  1  e.  CC
53, 4syl6eqel 2545 . . . 4  |-  ( x  =  1  ->  x  e.  CC )
6 id 22 . . . . 5  |-  ( x  =  -u 1  ->  x  =  -u 1 )
7 neg1cn 10523 . . . . 5  |-  -u 1  e.  CC
86, 7syl6eqel 2545 . . . 4  |-  ( x  =  -u 1  ->  x  e.  CC )
95, 8jaoi 379 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  e.  CC )
102, 9syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  e.  CC )
11 ax-1ne0 9449 . . . . . 6  |-  1  =/=  0
1211a1i 11 . . . . 5  |-  ( x  =  1  ->  1  =/=  0 )
133, 12eqnetrd 2739 . . . 4  |-  ( x  =  1  ->  x  =/=  0 )
14 neg1ne0 10525 . . . . . 6  |-  -u 1  =/=  0
1514a1i 11 . . . . 5  |-  ( x  =  -u 1  ->  -u 1  =/=  0 )
166, 15eqnetrd 2739 . . . 4  |-  ( x  =  -u 1  ->  x  =/=  0 )
1713, 16jaoi 379 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  =/=  0 )
182, 17syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  =/=  0 )
19 elpri 3992 . . 3  |-  ( y  e.  { 1 , 
-u 1 }  ->  ( y  =  1  \/  y  =  -u 1
) )
20 oveq12 6196 . . . . 5  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  =  ( 1  x.  1 ) )
214mulid1i 9486 . . . . . 6  |-  ( 1  x.  1 )  =  1
22 1ex 9479 . . . . . . 7  |-  1  e.  _V
2322prid1 4078 . . . . . 6  |-  1  e.  { 1 ,  -u
1 }
2421, 23eqeltri 2533 . . . . 5  |-  ( 1  x.  1 )  e. 
{ 1 ,  -u
1 }
2520, 24syl6eqel 2545 . . . 4  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
26 oveq12 6196 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  =  (
-u 1  x.  1 ) )
277mulid1i 9486 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
28 negex 9706 . . . . . . 7  |-  -u 1  e.  _V
2928prid2 4079 . . . . . 6  |-  -u 1  e.  { 1 ,  -u
1 }
3027, 29eqeltri 2533 . . . . 5  |-  ( -u
1  x.  1 )  e.  { 1 , 
-u 1 }
3126, 30syl6eqel 2545 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
32 oveq12 6196 . . . . 5  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  ( 1  x.  -u 1
) )
337mulid2i 9487 . . . . . 6  |-  ( 1  x.  -u 1 )  = 
-u 1
3433, 29eqeltri 2533 . . . . 5  |-  ( 1  x.  -u 1 )  e. 
{ 1 ,  -u
1 }
3532, 34syl6eqel 2545 . . . 4  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
36 oveq12 6196 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  (
-u 1  x.  -u 1
) )
37 neg1mulneg1e1 10637 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
3837, 23eqeltri 2533 . . . . 5  |-  ( -u
1  x.  -u 1
)  e.  { 1 ,  -u 1 }
3936, 38syl6eqel 2545 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
4025, 31, 35, 39ccase 937 . . 3  |-  ( ( ( x  =  1  \/  x  =  -u
1 )  /\  (
y  =  1  \/  y  =  -u 1
) )  ->  (
x  x.  y )  e.  { 1 , 
-u 1 } )
412, 19, 40syl2an 477 . 2  |-  ( ( x  e.  { 1 ,  -u 1 }  /\  y  e.  { 1 ,  -u 1 } )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
42 oveq2 6195 . . . . 5  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
43 1div1e1 10122 . . . . . 6  |-  ( 1  /  1 )  =  1
4443, 23eqeltri 2533 . . . . 5  |-  ( 1  /  1 )  e. 
{ 1 ,  -u
1 }
4542, 44syl6eqel 2545 . . . 4  |-  ( x  =  1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
46 oveq2 6195 . . . . 5  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
47 divneg2 10153 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
484, 4, 11, 47mp3an 1315 . . . . . . 7  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
4943negeqi 9701 . . . . . . 7  |-  -u (
1  /  1 )  =  -u 1
5048, 49eqtr3i 2481 . . . . . 6  |-  ( 1  /  -u 1 )  = 
-u 1
5150, 29eqeltri 2533 . . . . 5  |-  ( 1  /  -u 1 )  e. 
{ 1 ,  -u
1 }
5246, 51syl6eqel 2545 . . . 4  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
5345, 52jaoi 379 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  ( 1  /  x )  e. 
{ 1 ,  -u
1 } )
542, 53syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( 1  /  x )  e.  { 1 , 
-u 1 } )
551, 10, 18, 41, 23, 54cnmsubglem 17981 1  |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642    \ cdif 3420   {csn 3972   {cpr 3974   ` cfv 5513  (class class class)co 6187   CCcc 9378   0cc0 9380   1c1 9381    x. cmul 9385   -ucneg 9694    / cdiv 10091   ↾s cress 14274  SubGrpcsubg 15774  mulGrpcmgp 16693  ℂfldccnfld 17924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-addf 9459  ax-mulf 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-tpos 6842  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-fz 11536  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-sets 14279  df-ress 14280  df-plusg 14350  df-mulr 14351  df-starv 14352  df-tset 14356  df-ple 14357  df-ds 14359  df-unif 14360  df-0g 14479  df-mnd 15514  df-grp 15644  df-minusg 15645  df-subg 15777  df-cmn 16380  df-abl 16381  df-mgp 16694  df-ur 16706  df-rng 16750  df-cring 16751  df-oppr 16818  df-dvdsr 16836  df-unit 16837  df-invr 16867  df-dvr 16878  df-drng 16937  df-cnfld 17925
This theorem is referenced by:  cnmsgngrp  18115  psgninv  18118  zrhpsgnmhm  18120
  Copyright terms: Public domain W3C validator