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Theorem cnmsgnsubg 27411
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 3834 . . 3  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( x  =  1  \/  x  =  -u 1
) )
3 id 20 . . . . 5  |-  ( x  =  1  ->  x  =  1 )
4 ax-1cn 9048 . . . . 5  |-  1  e.  CC
53, 4syl6eqel 2524 . . . 4  |-  ( x  =  1  ->  x  e.  CC )
6 id 20 . . . . 5  |-  ( x  =  -u 1  ->  x  =  -u 1 )
7 neg1cn 10067 . . . . 5  |-  -u 1  e.  CC
86, 7syl6eqel 2524 . . . 4  |-  ( x  =  -u 1  ->  x  e.  CC )
95, 8jaoi 369 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  e.  CC )
102, 9syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  e.  CC )
11 ax-1ne0 9059 . . . . . 6  |-  1  =/=  0
1211a1i 11 . . . . 5  |-  ( x  =  1  ->  1  =/=  0 )
133, 12eqnetrd 2619 . . . 4  |-  ( x  =  1  ->  x  =/=  0 )
144, 11negne0i 9375 . . . . . 6  |-  -u 1  =/=  0
1514a1i 11 . . . . 5  |-  ( x  =  -u 1  ->  -u 1  =/=  0 )
166, 15eqnetrd 2619 . . . 4  |-  ( x  =  -u 1  ->  x  =/=  0 )
1713, 16jaoi 369 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  =/=  0 )
182, 17syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  =/=  0 )
19 elpri 3834 . . 3  |-  ( y  e.  { 1 , 
-u 1 }  ->  ( y  =  1  \/  y  =  -u 1
) )
20 oveq12 6090 . . . . 5  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  =  ( 1  x.  1 ) )
214mulid1i 9092 . . . . . 6  |-  ( 1  x.  1 )  =  1
22 1ex 9086 . . . . . . 7  |-  1  e.  _V
2322prid1 3912 . . . . . 6  |-  1  e.  { 1 ,  -u
1 }
2421, 23eqeltri 2506 . . . . 5  |-  ( 1  x.  1 )  e. 
{ 1 ,  -u
1 }
2520, 24syl6eqel 2524 . . . 4  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
26 oveq12 6090 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  =  (
-u 1  x.  1 ) )
277mulid1i 9092 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
28 negex 9304 . . . . . . 7  |-  -u 1  e.  _V
2928prid2 3913 . . . . . 6  |-  -u 1  e.  { 1 ,  -u
1 }
3027, 29eqeltri 2506 . . . . 5  |-  ( -u
1  x.  1 )  e.  { 1 , 
-u 1 }
3126, 30syl6eqel 2524 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
32 oveq12 6090 . . . . 5  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  ( 1  x.  -u 1
) )
337mulid2i 9093 . . . . . 6  |-  ( 1  x.  -u 1 )  = 
-u 1
3433, 29eqeltri 2506 . . . . 5  |-  ( 1  x.  -u 1 )  e. 
{ 1 ,  -u
1 }
3532, 34syl6eqel 2524 . . . 4  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
36 oveq12 6090 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  (
-u 1  x.  -u 1
) )
374, 4mul2negi 9481 . . . . . . 7  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
3837, 21eqtri 2456 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
3938, 23eqeltri 2506 . . . . 5  |-  ( -u
1  x.  -u 1
)  e.  { 1 ,  -u 1 }
4036, 39syl6eqel 2524 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
4125, 31, 35, 40ccase 913 . . 3  |-  ( ( ( x  =  1  \/  x  =  -u
1 )  /\  (
y  =  1  \/  y  =  -u 1
) )  ->  (
x  x.  y )  e.  { 1 , 
-u 1 } )
422, 19, 41syl2an 464 . 2  |-  ( ( x  e.  { 1 ,  -u 1 }  /\  y  e.  { 1 ,  -u 1 } )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
43 oveq2 6089 . . . . 5  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
444div1i 9742 . . . . . 6  |-  ( 1  /  1 )  =  1
4544, 23eqeltri 2506 . . . . 5  |-  ( 1  /  1 )  e. 
{ 1 ,  -u
1 }
4643, 45syl6eqel 2524 . . . 4  |-  ( x  =  1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
47 oveq2 6089 . . . . 5  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
48 divneg2 9738 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
494, 4, 11, 48mp3an 1279 . . . . . . 7  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
5044negeqi 9299 . . . . . . 7  |-  -u (
1  /  1 )  =  -u 1
5149, 50eqtr3i 2458 . . . . . 6  |-  ( 1  /  -u 1 )  = 
-u 1
5251, 29eqeltri 2506 . . . . 5  |-  ( 1  /  -u 1 )  e. 
{ 1 ,  -u
1 }
5347, 52syl6eqel 2524 . . . 4  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
5446, 53jaoi 369 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  ( 1  /  x )  e. 
{ 1 ,  -u
1 } )
552, 54syl 16 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( 1  /  x )  e.  { 1 , 
-u 1 } )
561, 10, 18, 42, 23, 55cnmsubglem 16761 1  |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {csn 3814   {cpr 3815   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995   -ucneg 9292    / cdiv 9677   ↾s cress 13470  SubGrpcsubg 14938  mulGrpcmgp 15648  ℂfldccnfld 16703
This theorem is referenced by:  cnmsgngrp  27413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-cnfld 16704
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