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Theorem cnmsgnsubg 19222
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 3976 . . 3  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( x  =  1  \/  x  =  -u 1
) )
3 id 22 . . . . 5  |-  ( x  =  1  ->  x  =  1 )
4 ax-1cn 9615 . . . . 5  |-  1  e.  CC
53, 4syl6eqel 2557 . . . 4  |-  ( x  =  1  ->  x  e.  CC )
6 id 22 . . . . 5  |-  ( x  =  -u 1  ->  x  =  -u 1 )
7 neg1cn 10735 . . . . 5  |-  -u 1  e.  CC
86, 7syl6eqel 2557 . . . 4  |-  ( x  =  -u 1  ->  x  e.  CC )
95, 8jaoi 386 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  e.  CC )
102, 9syl 17 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  e.  CC )
11 ax-1ne0 9626 . . . . . 6  |-  1  =/=  0
1211a1i 11 . . . . 5  |-  ( x  =  1  ->  1  =/=  0 )
133, 12eqnetrd 2710 . . . 4  |-  ( x  =  1  ->  x  =/=  0 )
14 neg1ne0 10737 . . . . . 6  |-  -u 1  =/=  0
1514a1i 11 . . . . 5  |-  ( x  =  -u 1  ->  -u 1  =/=  0 )
166, 15eqnetrd 2710 . . . 4  |-  ( x  =  -u 1  ->  x  =/=  0 )
1713, 16jaoi 386 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  =/=  0 )
182, 17syl 17 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  =/=  0 )
19 elpri 3976 . . 3  |-  ( y  e.  { 1 , 
-u 1 }  ->  ( y  =  1  \/  y  =  -u 1
) )
20 oveq12 6317 . . . . 5  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  =  ( 1  x.  1 ) )
214mulid1i 9663 . . . . . 6  |-  ( 1  x.  1 )  =  1
22 1ex 9656 . . . . . . 7  |-  1  e.  _V
2322prid1 4071 . . . . . 6  |-  1  e.  { 1 ,  -u
1 }
2421, 23eqeltri 2545 . . . . 5  |-  ( 1  x.  1 )  e. 
{ 1 ,  -u
1 }
2520, 24syl6eqel 2557 . . . 4  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
26 oveq12 6317 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  =  (
-u 1  x.  1 ) )
277mulid1i 9663 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
28 negex 9893 . . . . . . 7  |-  -u 1  e.  _V
2928prid2 4072 . . . . . 6  |-  -u 1  e.  { 1 ,  -u
1 }
3027, 29eqeltri 2545 . . . . 5  |-  ( -u
1  x.  1 )  e.  { 1 , 
-u 1 }
3126, 30syl6eqel 2557 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
32 oveq12 6317 . . . . 5  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  ( 1  x.  -u 1
) )
337mulid2i 9664 . . . . . 6  |-  ( 1  x.  -u 1 )  = 
-u 1
3433, 29eqeltri 2545 . . . . 5  |-  ( 1  x.  -u 1 )  e. 
{ 1 ,  -u
1 }
3532, 34syl6eqel 2557 . . . 4  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
36 oveq12 6317 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  (
-u 1  x.  -u 1
) )
37 neg1mulneg1e1 10850 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
3837, 23eqeltri 2545 . . . . 5  |-  ( -u
1  x.  -u 1
)  e.  { 1 ,  -u 1 }
3936, 38syl6eqel 2557 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
4025, 31, 35, 39ccase 961 . . 3  |-  ( ( ( x  =  1  \/  x  =  -u
1 )  /\  (
y  =  1  \/  y  =  -u 1
) )  ->  (
x  x.  y )  e.  { 1 , 
-u 1 } )
412, 19, 40syl2an 485 . 2  |-  ( ( x  e.  { 1 ,  -u 1 }  /\  y  e.  { 1 ,  -u 1 } )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
42 oveq2 6316 . . . . 5  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
43 1div1e1 10322 . . . . . 6  |-  ( 1  /  1 )  =  1
4443, 23eqeltri 2545 . . . . 5  |-  ( 1  /  1 )  e. 
{ 1 ,  -u
1 }
4542, 44syl6eqel 2557 . . . 4  |-  ( x  =  1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
46 oveq2 6316 . . . . 5  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
47 divneg2 10353 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
484, 4, 11, 47mp3an 1390 . . . . . . 7  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
4943negeqi 9888 . . . . . . 7  |-  -u (
1  /  1 )  =  -u 1
5048, 49eqtr3i 2495 . . . . . 6  |-  ( 1  /  -u 1 )  = 
-u 1
5150, 29eqeltri 2545 . . . . 5  |-  ( 1  /  -u 1 )  e. 
{ 1 ,  -u
1 }
5246, 51syl6eqel 2557 . . . 4  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
5345, 52jaoi 386 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  ( 1  /  x )  e. 
{ 1 ,  -u
1 } )
542, 53syl 17 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( 1  /  x )  e.  { 1 , 
-u 1 } )
551, 10, 18, 41, 23, 54cnmsubglem 19107 1  |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    x. cmul 9562   -ucneg 9881    / cdiv 10291   ↾s cress 15200  SubGrpcsubg 16889  mulGrpcmgp 17801  ℂfldccnfld 19047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-subg 16892  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-cnfld 19048
This theorem is referenced by:  cnmsgngrp  19224  psgninv  19227  zrhpsgnmhm  19229
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