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Theorem readdsubgo 25031
Description: The real numbers under addition comprise a subgroup of the complex numbers under addition. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)
Assertion
Ref Expression
readdsubgo  |-  (  +  |`  ( RR  X.  RR ) )  e.  (
SubGrpOp `  +  )

Proof of Theorem readdsubgo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnaddablo 25028 . . 3  |-  +  e.  AbelOp
2 ablogrpo 24962 . . 3  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . 2  |-  +  e.  GrpOp
4 ax-addf 9567 . . . 4  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5734 . . 3  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 24890 . 2  |-  CC  =  ran  +
7 cnid 25029 . 2  |-  0  =  (GId `  +  )
8 eqid 2467 . 2  |-  ( inv `  +  )  =  ( inv `  +  )
9 ax-resscn 9545 . 2  |-  RR  C_  CC
10 eqid 2467 . 2  |-  (  +  |`  ( RR  X.  RR ) )  =  (  +  |`  ( RR  X.  RR ) )
11 readdcl 9571 . 2  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
12 0re 9592 . 2  |-  0  e.  RR
13 recn 9578 . . . 4  |-  ( x  e.  RR  ->  x  e.  CC )
14 addinv 25030 . . . 4  |-  ( x  e.  CC  ->  (
( inv `  +  ) `  x )  =  -u x )
1513, 14syl 16 . . 3  |-  ( x  e.  RR  ->  (
( inv `  +  ) `  x )  =  -u x )
16 renegcl 9878 . . 3  |-  ( x  e.  RR  ->  -u x  e.  RR )
1715, 16eqeltrd 2555 . 2  |-  ( x  e.  RR  ->  (
( inv `  +  ) `  x )  e.  RR )
183, 6, 7, 8, 9, 10, 11, 12, 17issubgoi 24988 1  |-  (  +  |`  ( RR  X.  RR ) )  e.  (
SubGrpOp `  +  )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767    X. cxp 4997    |` cres 5001   ` cfv 5586   CCcc 9486   RRcr 9487   0cc0 9488    + caddc 9491   -ucneg 9802   GrpOpcgr 24864   invcgn 24866   AbelOpcablo 24959   SubGrpOpcsubgo 24979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629  df-sub 9803  df-neg 9804  df-grpo 24869  df-gid 24870  df-ginv 24871  df-ablo 24960  df-subgo 24980
This theorem is referenced by:  circgrp  25052
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