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Theorem readdsubgo 21941
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 21938 . . 3  |-  +  e.  AbelOp
2 ablogrpo 21872 . . 3  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 8 . 2  |-  +  e.  GrpOp
4 ax-addf 9069 . . . 4  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5596 . . 3  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 21800 . 2  |-  CC  =  ran  +
7 cnid 21939 . 2  |-  0  =  (GId `  +  )
8 eqid 2436 . 2  |-  ( inv `  +  )  =  ( inv `  +  )
9 ax-resscn 9047 . 2  |-  RR  C_  CC
10 eqid 2436 . 2  |-  (  +  |`  ( RR  X.  RR ) )  =  (  +  |`  ( RR  X.  RR ) )
11 readdcl 9073 . 2  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
12 0re 9091 . 2  |-  0  e.  RR
13 recn 9080 . . . 4  |-  ( x  e.  RR  ->  x  e.  CC )
14 addinv 21940 . . . 4  |-  ( x  e.  CC  ->  (
( inv `  +  ) `  x )  =  -u x )
1513, 14syl 16 . . 3  |-  ( x  e.  RR  ->  (
( inv `  +  ) `  x )  =  -u x )
16 renegcl 9364 . . 3  |-  ( x  e.  RR  ->  -u x  e.  RR )
1715, 16eqeltrd 2510 . 2  |-  ( x  e.  RR  ->  (
( inv `  +  ) `  x )  e.  RR )
183, 6, 7, 8, 9, 10, 11, 12, 17issubgoi 21898 1  |-  (  +  |`  ( RR  X.  RR ) )  e.  (
SubGrpOp `  +  )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    X. cxp 4876    |` cres 4880   ` cfv 5454   CCcc 8988   RRcr 8989   0cc0 8990    + caddc 8993   -ucneg 9292   GrpOpcgr 21774   invcgn 21776   AbelOpcablo 21869   SubGrpOpcsubgo 21889
This theorem is referenced by:  circgrp  21962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-grpo 21779  df-gid 21780  df-ginv 21781  df-ablo 21870  df-subgo 21890
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