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Theorem cnmsgnsubg 26757
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 3736 . . 3  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( x  =  1  \/  x  =  -u 1
) )
3 id 19 . . . . 5  |-  ( x  =  1  ->  x  =  1 )
4 ax-1cn 8882 . . . . 5  |-  1  e.  CC
53, 4syl6eqel 2446 . . . 4  |-  ( x  =  1  ->  x  e.  CC )
6 id 19 . . . . 5  |-  ( x  =  -u 1  ->  x  =  -u 1 )
7 neg1cn 9900 . . . . 5  |-  -u 1  e.  CC
86, 7syl6eqel 2446 . . . 4  |-  ( x  =  -u 1  ->  x  e.  CC )
95, 8jaoi 368 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  e.  CC )
102, 9syl 15 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  e.  CC )
11 ax-1ne0 8893 . . . . . 6  |-  1  =/=  0
1211a1i 10 . . . . 5  |-  ( x  =  1  ->  1  =/=  0 )
133, 12eqnetrd 2539 . . . 4  |-  ( x  =  1  ->  x  =/=  0 )
144, 11negne0i 9208 . . . . . 6  |-  -u 1  =/=  0
1514a1i 10 . . . . 5  |-  ( x  =  -u 1  ->  -u 1  =/=  0 )
166, 15eqnetrd 2539 . . . 4  |-  ( x  =  -u 1  ->  x  =/=  0 )
1713, 16jaoi 368 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  =/=  0 )
182, 17syl 15 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  x  =/=  0 )
19 elpri 3736 . . 3  |-  ( y  e.  { 1 , 
-u 1 }  ->  ( y  =  1  \/  y  =  -u 1
) )
20 oveq12 5951 . . . . 5  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  =  ( 1  x.  1 ) )
214mulid1i 8926 . . . . . 6  |-  ( 1  x.  1 )  =  1
22 1ex 8920 . . . . . . 7  |-  1  e.  _V
2322prid1 3810 . . . . . 6  |-  1  e.  { 1 ,  -u
1 }
2421, 23eqeltri 2428 . . . . 5  |-  ( 1  x.  1 )  e. 
{ 1 ,  -u
1 }
2520, 24syl6eqel 2446 . . . 4  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
26 oveq12 5951 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  =  (
-u 1  x.  1 ) )
277mulid1i 8926 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
28 negex 9137 . . . . . . 7  |-  -u 1  e.  _V
2928prid2 3811 . . . . . 6  |-  -u 1  e.  { 1 ,  -u
1 }
3027, 29eqeltri 2428 . . . . 5  |-  ( -u
1  x.  1 )  e.  { 1 , 
-u 1 }
3126, 30syl6eqel 2446 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
32 oveq12 5951 . . . . 5  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  ( 1  x.  -u 1
) )
337mulid2i 8927 . . . . . 6  |-  ( 1  x.  -u 1 )  = 
-u 1
3433, 29eqeltri 2428 . . . . 5  |-  ( 1  x.  -u 1 )  e. 
{ 1 ,  -u
1 }
3532, 34syl6eqel 2446 . . . 4  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
36 oveq12 5951 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  (
-u 1  x.  -u 1
) )
374, 4mul2negi 9314 . . . . . . 7  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
3837, 21eqtri 2378 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
3938, 23eqeltri 2428 . . . . 5  |-  ( -u
1  x.  -u 1
)  e.  { 1 ,  -u 1 }
4036, 39syl6eqel 2446 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
4125, 31, 35, 40ccase 912 . . 3  |-  ( ( ( x  =  1  \/  x  =  -u
1 )  /\  (
y  =  1  \/  y  =  -u 1
) )  ->  (
x  x.  y )  e.  { 1 , 
-u 1 } )
422, 19, 41syl2an 463 . 2  |-  ( ( x  e.  { 1 ,  -u 1 }  /\  y  e.  { 1 ,  -u 1 } )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
43 oveq2 5950 . . . . 5  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
444div1i 9575 . . . . . 6  |-  ( 1  /  1 )  =  1
4544, 23eqeltri 2428 . . . . 5  |-  ( 1  /  1 )  e. 
{ 1 ,  -u
1 }
4643, 45syl6eqel 2446 . . . 4  |-  ( x  =  1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
47 oveq2 5950 . . . . 5  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
48 divneg2 9571 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
494, 4, 11, 48mp3an 1277 . . . . . . 7  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
5044negeqi 9132 . . . . . . 7  |-  -u (
1  /  1 )  =  -u 1
5149, 50eqtr3i 2380 . . . . . 6  |-  ( 1  /  -u 1 )  = 
-u 1
5251, 29eqeltri 2428 . . . . 5  |-  ( 1  /  -u 1 )  e. 
{ 1 ,  -u
1 }
5347, 52syl6eqel 2446 . . . 4  |-  ( x  =  -u 1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
5446, 53jaoi 368 . . 3  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  ( 1  /  x )  e. 
{ 1 ,  -u
1 } )
552, 54syl 15 . 2  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( 1  /  x )  e.  { 1 , 
-u 1 } )
561, 10, 18, 42, 23, 55cnmsubglem 16534 1  |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225   {csn 3716   {cpr 3717   ` cfv 5334  (class class class)co 5942   CCcc 8822   0cc0 8824   1c1 8825    x. cmul 8829   -ucneg 9125    / cdiv 9510   ↾s cress 13240  SubGrpcsubg 14708  mulGrpcmgp 15418  ℂfldccnfld 16476
This theorem is referenced by:  cnmsgngrp  26759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-tpos 6318  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-fz 10872  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-0g 13497  df-mnd 14460  df-grp 14582  df-minusg 14583  df-subg 14711  df-cmn 15184  df-abl 15185  df-mgp 15419  df-rng 15433  df-cring 15434  df-ur 15435  df-oppr 15498  df-dvdsr 15516  df-unit 15517  df-invr 15547  df-dvr 15558  df-drng 15607  df-cnfld 16477
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