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Theorem readdsubgo 21073
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 21070 . . 3  |-  +  e.  AbelOp
2 ablogrpo 21004 . . 3  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 8 . 2  |-  +  e.  GrpOp
4 ax-addf 8861 . . . 4  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5432 . . 3  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 20932 . 2  |-  CC  =  ran  +
7 cnid 21071 . 2  |-  0  =  (GId `  +  )
8 eqid 2316 . 2  |-  ( inv `  +  )  =  ( inv `  +  )
9 ax-resscn 8839 . 2  |-  RR  C_  CC
10 eqid 2316 . 2  |-  (  +  |`  ( RR  X.  RR ) )  =  (  +  |`  ( RR  X.  RR ) )
11 readdcl 8865 . 2  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
12 0re 8883 . 2  |-  0  e.  RR
13 recn 8872 . . . 4  |-  ( x  e.  RR  ->  x  e.  CC )
14 addinv 21072 . . . 4  |-  ( x  e.  CC  ->  (
( inv `  +  ) `  x )  =  -u x )
1513, 14syl 15 . . 3  |-  ( x  e.  RR  ->  (
( inv `  +  ) `  x )  =  -u x )
16 renegcl 9155 . . 3  |-  ( x  e.  RR  ->  -u x  e.  RR )
1715, 16eqeltrd 2390 . 2  |-  ( x  e.  RR  ->  (
( inv `  +  ) `  x )  e.  RR )
183, 6, 7, 8, 9, 10, 11, 12, 17issubgoi 21030 1  |-  (  +  |`  ( RR  X.  RR ) )  e.  (
SubGrpOp `  +  )
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701    X. cxp 4724    |` cres 4728   ` cfv 5292   CCcc 8780   RRcr 8781   0cc0 8782    + caddc 8785   -ucneg 9083   GrpOpcgr 20906   invcgn 20908   AbelOpcablo 21001   SubGrpOpcsubgo 21021
This theorem is referenced by:  circgrp  21094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-addf 8861
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-ltxr 8917  df-sub 9084  df-neg 9085  df-grpo 20911  df-gid 20912  df-ginv 20913  df-ablo 21002  df-subgo 21022
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