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Theorem addinv 26080
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 26078 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 26012 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9618 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5734 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 25940 . . . 4  |-  CC  =  ran  +
7 cnid 26079 . . . 4  |-  0  =  (GId `  +  )
8 eqid 2451 . . . 4  |-  ( inv `  +  )  =  ( inv `  +  )
96, 7, 8grpoinvval 25953 . . 3  |-  ( (  +  e.  GrpOp  /\  A  e.  CC )  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
103, 9mpan 676 . 2  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
11 df-neg 9863 . . . . 5  |-  -u A  =  ( 0  -  A )
1211oveq1i 6300 . . . 4  |-  ( -u A  +  A )  =  ( ( 0  -  A )  +  A )
13 0cn 9635 . . . . 5  |-  0  e.  CC
14 npcan 9884 . . . . 5  |-  ( ( 0  e.  CC  /\  A  e.  CC )  ->  ( ( 0  -  A )  +  A
)  =  0 )
1513, 14mpan 676 . . . 4  |-  ( A  e.  CC  ->  (
( 0  -  A
)  +  A )  =  0 )
1612, 15syl5eq 2497 . . 3  |-  ( A  e.  CC  ->  ( -u A  +  A )  =  0 )
17 negcl 9875 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
18 negeu 9865 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  E! y  e.  CC  ( A  +  y
)  =  0 )
1913, 18mpan2 677 . . . . 5  |-  ( A  e.  CC  ->  E! y  e.  CC  ( A  +  y )  =  0 )
20 addcom 9819 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  +  y )  =  ( y  +  A ) )
2120eqeq1d 2453 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  =  0  <-> 
( y  +  A
)  =  0 ) )
2221reubidva 2974 . . . . 5  |-  ( A  e.  CC  ->  ( E! y  e.  CC  ( A  +  y
)  =  0  <->  E! y  e.  CC  (
y  +  A )  =  0 ) )
2319, 22mpbid 214 . . . 4  |-  ( A  e.  CC  ->  E! y  e.  CC  (
y  +  A )  =  0 )
24 oveq1 6297 . . . . . 6  |-  ( y  =  -u A  ->  (
y  +  A )  =  ( -u A  +  A ) )
2524eqeq1d 2453 . . . . 5  |-  ( y  =  -u A  ->  (
( y  +  A
)  =  0  <->  ( -u A  +  A )  =  0 ) )
2625riota2 6274 . . . 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 667 . . 3  |-  ( A  e.  CC  ->  (
( -u A  +  A
)  =  0  <->  ( iota_ y  e.  CC  (
y  +  A )  =  0 )  = 
-u A ) )
2816, 27mpbid 214 . 2  |-  ( A  e.  CC  ->  ( iota_ y  e.  CC  (
y  +  A )  =  0 )  = 
-u A )
2910, 28eqtrd 2485 1  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  -u A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E!wreu 2739    X. cxp 4832   ` cfv 5582   iota_crio 6251  (class class class)co 6290   CCcc 9537   0cc0 9539    + caddc 9542    - cmin 9860   -ucneg 9861   GrpOpcgr 25914   invcgn 25916   AbelOpcablo 26009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-addf 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-ltxr 9680  df-sub 9862  df-neg 9863  df-grpo 25919  df-gid 25920  df-ginv 25921  df-ablo 26010
This theorem is referenced by:  readdsubgo  26081  zaddsubgo  26082
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