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Theorem cnmsgnsubg 18786
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 4036 . . 3  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( x  =  1  \/  x  =  -u 1
) )
3 id 22 . . . . 5  |-  ( x  =  1  ->  x  =  1 )
4 ax-1cn 9539 . . . . 5  |-  1  e.  CC
53, 4syl6eqel 2550 . . . 4  |-  ( x  =  1  ->  x  e.  CC )
6 id 22 . . . . 5  |-  ( x  =  -u 1  ->  x  =  -u 1 )
7 neg1cn 10635 . . . . 5  |-  -u 1  e.  CC
86, 7syl6eqel 2550 . . . 4  |-  ( x  =  -u 1  ->  x  e.  CC )
95, 8jaoi 377 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  e.  CC )
102, 9syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  e.  CC )
11 ax-1ne0 9550 . . . . . 6  |-  1  =/=  0
1211a1i 11 . . . . 5  |-  ( x  =  1  ->  1  =/=  0 )
133, 12eqnetrd 2747 . . . 4  |-  ( x  =  1  ->  x  =/=  0 )
14 neg1ne0 10637 . . . . . 6  |-  -u 1  =/=  0
1514a1i 11 . . . . 5  |-  ( x  =  -u 1  ->  -u 1  =/=  0 )
166, 15eqnetrd 2747 . . . 4  |-  ( x  =  -u 1  ->  x  =/=  0 )
1713, 16jaoi 377 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  =/=  0 )
182, 17syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  =/=  0 )
19 elpri 4036 . . 3  |-  ( y  e.  { 1 , 
-u 1 }  ->  ( y  =  1  \/  y  =  -u 1
) )
20 oveq12 6279 . . . . 5  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  =  ( 1  x.  1 ) )
214mulid1i 9587 . . . . . 6  |-  ( 1  x.  1 )  =  1
22 1ex 9580 . . . . . . 7  |-  1  e.  _V
2322prid1 4124 . . . . . 6  |-  1  e.  { 1 ,  -u
1 }
2421, 23eqeltri 2538 . . . . 5  |-  ( 1  x.  1 )  e. 
{ 1 ,  -u
1 }
2520, 24syl6eqel 2550 . . . 4  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
26 oveq12 6279 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  =  (
-u 1  x.  1 ) )
277mulid1i 9587 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
28 negex 9809 . . . . . . 7  |-  -u 1  e.  _V
2928prid2 4125 . . . . . 6  |-  -u 1  e.  { 1 ,  -u
1 }
3027, 29eqeltri 2538 . . . . 5  |-  ( -u
1  x.  1 )  e.  { 1 , 
-u 1 }
3126, 30syl6eqel 2550 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
32 oveq12 6279 . . . . 5  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  ( 1  x.  -u 1
) )
337mulid2i 9588 . . . . . 6  |-  ( 1  x.  -u 1 )  = 
-u 1
3433, 29eqeltri 2538 . . . . 5  |-  ( 1  x.  -u 1 )  e. 
{ 1 ,  -u
1 }
3532, 34syl6eqel 2550 . . . 4  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
36 oveq12 6279 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  (
-u 1  x.  -u 1
) )
37 neg1mulneg1e1 10749 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
3837, 23eqeltri 2538 . . . . 5  |-  ( -u
1  x.  -u 1
)  e.  { 1 ,  -u 1 }
3936, 38syl6eqel 2550 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
4025, 31, 35, 39ccase 944 . . 3  |-  ( ( ( x  =  1  \/  x  =  -u
1 )  /\  (
y  =  1  \/  y  =  -u 1
) )  ->  (
x  x.  y )  e.  { 1 , 
-u 1 } )
412, 19, 40syl2an 475 . 2  |-  ( ( x  e.  { 1 ,  -u 1 }  /\  y  e.  { 1 ,  -u 1 } )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
42 oveq2 6278 . . . . 5  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
43 1div1e1 10233 . . . . . 6  |-  ( 1  /  1 )  =  1
4443, 23eqeltri 2538 . . . . 5  |-  ( 1  /  1 )  e. 
{ 1 ,  -u
1 }
4542, 44syl6eqel 2550 . . . 4  |-  ( x  =  1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
46 oveq2 6278 . . . . 5  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
47 divneg2 10264 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
484, 4, 11, 47mp3an 1322 . . . . . . 7  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
4943negeqi 9804 . . . . . . 7  |-  -u (
1  /  1 )  =  -u 1
5048, 49eqtr3i 2485 . . . . . 6  |-  ( 1  /  -u 1 )  = 
-u 1
5150, 29eqeltri 2538 . . . . 5  |-  ( 1  /  -u 1 )  e. 
{ 1 ,  -u
1 }
5246, 51syl6eqel 2550 . . . 4  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
5345, 52jaoi 377 . . 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 18675 1  |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458   {csn 4016   {cpr 4018   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486   -ucneg 9797    / cdiv 10202   ↾s cress 14717  SubGrpcsubg 16394  mulGrpcmgp 17336  ℂfldccnfld 18615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-subg 16397  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-drng 17593  df-cnfld 18616
This theorem is referenced by:  cnmsgngrp  18788  psgninv  18791  zrhpsgnmhm  18793
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