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Theorem cnmsgnsubg 27434
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 3660 . . 3  |-  ( x  e.  { 1 , 
-u 1 }  ->  ( x  =  1  \/  x  =  -u 1
) )
3 id 19 . . . . 5  |-  ( x  =  1  ->  x  =  1 )
4 ax-1cn 8795 . . . . 5  |-  1  e.  CC
53, 4syl6eqel 2371 . . . 4  |-  ( x  =  1  ->  x  e.  CC )
6 id 19 . . . . 5  |-  ( x  =  -u 1  ->  x  =  -u 1 )
7 neg1cn 9813 . . . . 5  |-  -u 1  e.  CC
86, 7syl6eqel 2371 . . . 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 8806 . . . . . 6  |-  1  =/=  0
1211a1i 10 . . . . 5  |-  ( x  =  1  ->  1  =/=  0 )
133, 12eqnetrd 2464 . . . 4  |-  ( x  =  1  ->  x  =/=  0 )
144, 11negne0i 9121 . . . . . 6  |-  -u 1  =/=  0
1514a1i 10 . . . . 5  |-  ( x  =  -u 1  ->  -u 1  =/=  0 )
166, 15eqnetrd 2464 . . . 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 3660 . . 3  |-  ( y  e.  { 1 , 
-u 1 }  ->  ( y  =  1  \/  y  =  -u 1
) )
20 oveq12 5867 . . . . 5  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  =  ( 1  x.  1 ) )
214mulid1i 8839 . . . . . 6  |-  ( 1  x.  1 )  =  1
22 1ex 8833 . . . . . . 7  |-  1  e.  _V
2322prid1 3734 . . . . . 6  |-  1  e.  { 1 ,  -u
1 }
2421, 23eqeltri 2353 . . . . 5  |-  ( 1  x.  1 )  e. 
{ 1 ,  -u
1 }
2520, 24syl6eqel 2371 . . . 4  |-  ( ( x  =  1  /\  y  =  1 )  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
26 oveq12 5867 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  =  (
-u 1  x.  1 ) )
277mulid1i 8839 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
28 negex 9050 . . . . . . 7  |-  -u 1  e.  _V
2928prid2 3735 . . . . . 6  |-  -u 1  e.  { 1 ,  -u
1 }
3027, 29eqeltri 2353 . . . . 5  |-  ( -u
1  x.  1 )  e.  { 1 , 
-u 1 }
3126, 30syl6eqel 2371 . . . 4  |-  ( ( x  =  -u 1  /\  y  =  1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
32 oveq12 5867 . . . . 5  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  ( 1  x.  -u 1
) )
337mulid2i 8840 . . . . . 6  |-  ( 1  x.  -u 1 )  = 
-u 1
3433, 29eqeltri 2353 . . . . 5  |-  ( 1  x.  -u 1 )  e. 
{ 1 ,  -u
1 }
3532, 34syl6eqel 2371 . . . 4  |-  ( ( x  =  1  /\  y  =  -u 1
)  ->  ( x  x.  y )  e.  {
1 ,  -u 1 } )
36 oveq12 5867 . . . . 5  |-  ( ( x  =  -u 1  /\  y  =  -u 1
)  ->  ( x  x.  y )  =  (
-u 1  x.  -u 1
) )
374, 4mul2negi 9227 . . . . . . 7  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
3837, 21eqtri 2303 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
3938, 23eqeltri 2353 . . . . 5  |-  ( -u
1  x.  -u 1
)  e.  { 1 ,  -u 1 }
4036, 39syl6eqel 2371 . . . 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 5866 . . . . 5  |-  ( x  =  1  ->  (
1  /  x )  =  ( 1  / 
1 ) )
444div1i 9488 . . . . . 6  |-  ( 1  /  1 )  =  1
4544, 23eqeltri 2353 . . . . 5  |-  ( 1  /  1 )  e. 
{ 1 ,  -u
1 }
4643, 45syl6eqel 2371 . . . 4  |-  ( x  =  1  ->  (
1  /  x )  e.  { 1 , 
-u 1 } )
47 oveq2 5866 . . . . 5  |-  ( x  =  -u 1  ->  (
1  /  x )  =  ( 1  /  -u 1 ) )
48 divneg2 9484 . . . . . . . 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 9045 . . . . . . 7  |-  -u (
1  /  1 )  =  -u 1
5149, 50eqtr3i 2305 . . . . . 6  |-  ( 1  /  -u 1 )  = 
-u 1
5251, 29eqeltri 2353 . . . . 5  |-  ( 1  /  -u 1 )  e. 
{ 1 ,  -u
1 }
5347, 52syl6eqel 2371 . . . 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 16434 1  |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   -ucneg 9038    / cdiv 9423   ↾s cress 13149  SubGrpcsubg 14615  mulGrpcmgp 15325  ℂfldccnfld 16377
This theorem is referenced by:  cnmsgngrp  27436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-cnfld 16378
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