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Theorem addinv 23851
Description: Value of the group inverse of complex number addition. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addinv  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  -u A )

Proof of Theorem addinv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnaddablo 23849 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 23783 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9373 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5576 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 23711 . . . 4  |-  CC  =  ran  +
7 cnid 23850 . . . 4  |-  0  =  (GId `  +  )
8 eqid 2443 . . . 4  |-  ( inv `  +  )  =  ( inv `  +  )
96, 7, 8grpoinvval 23724 . . 3  |-  ( (  +  e.  GrpOp  /\  A  e.  CC )  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
103, 9mpan 670 . 2  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
11 df-neg 9610 . . . . 5  |-  -u A  =  ( 0  -  A )
1211oveq1i 6113 . . . 4  |-  ( -u A  +  A )  =  ( ( 0  -  A )  +  A )
13 0cn 9390 . . . . 5  |-  0  e.  CC
14 npcan 9631 . . . . 5  |-  ( ( 0  e.  CC  /\  A  e.  CC )  ->  ( ( 0  -  A )  +  A
)  =  0 )
1513, 14mpan 670 . . . 4  |-  ( A  e.  CC  ->  (
( 0  -  A
)  +  A )  =  0 )
1612, 15syl5eq 2487 . . 3  |-  ( A  e.  CC  ->  ( -u A  +  A )  =  0 )
17 negcl 9622 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
18 negeu 9612 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  E! y  e.  CC  ( A  +  y
)  =  0 )
1913, 18mpan2 671 . . . . 5  |-  ( A  e.  CC  ->  E! y  e.  CC  ( A  +  y )  =  0 )
20 addcom 9567 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  +  y )  =  ( y  +  A ) )
2120eqeq1d 2451 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  =  0  <-> 
( y  +  A
)  =  0 ) )
2221reubidva 2916 . . . . 5  |-  ( A  e.  CC  ->  ( E! y  e.  CC  ( A  +  y
)  =  0  <->  E! y  e.  CC  (
y  +  A )  =  0 ) )
2319, 22mpbid 210 . . . 4  |-  ( A  e.  CC  ->  E! y  e.  CC  (
y  +  A )  =  0 )
24 oveq1 6110 . . . . . 6  |-  ( y  =  -u A  ->  (
y  +  A )  =  ( -u A  +  A ) )
2524eqeq1d 2451 . . . . 5  |-  ( y  =  -u A  ->  (
( y  +  A
)  =  0  <->  ( -u A  +  A )  =  0 ) )
2625riota2 6087 . . . 4  |-  ( (
-u A  e.  CC  /\  E! y  e.  CC  ( y  +  A
)  =  0 )  ->  ( ( -u A  +  A )  =  0  <->  ( iota_ y  e.  CC  ( y  +  A )  =  0 )  =  -u A ) )
2717, 23, 26syl2anc 661 . . 3  |-  ( A  e.  CC  ->  (
( -u A  +  A
)  =  0  <->  ( iota_ y  e.  CC  (
y  +  A )  =  0 )  = 
-u A ) )
2816, 27mpbid 210 . 2  |-  ( A  e.  CC  ->  ( iota_ y  e.  CC  (
y  +  A )  =  0 )  = 
-u A )
2910, 28eqtrd 2475 1  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  -u A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E!wreu 2729    X. cxp 4850   ` cfv 5430   iota_crio 6063  (class class class)co 6103   CCcc 9292   0cc0 9294    + caddc 9297    - cmin 9607   -ucneg 9608   GrpOpcgr 23685   invcgn 23687   AbelOpcablo 23780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-addf 9373
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-ltxr 9435  df-sub 9609  df-neg 9610  df-grpo 23690  df-gid 23691  df-ginv 23692  df-ablo 23781
This theorem is referenced by:  readdsubgo  23852  zaddsubgo  23853
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