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Theorem rngodmeqrn 25419
Description: In a unital ring the domain of the first operand of the addition equals the domain of the second operand of the addition. (Contributed by FL, 11-Feb-2010.)
Hypothesis
Ref Expression
rnplrnml3.1  |-  G  =  ( 1st `  R
)
Assertion
Ref Expression
rngodmeqrn  |-  ( R  e.  RingOps  ->  dom  dom  G  =  ran  dom  G )

Proof of Theorem rngodmeqrn
StepHypRef Expression
1 rnplrnml3.1 . . 3  |-  G  =  ( 1st `  R
)
21rngogrpo 21057 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 eqid 2283 . . . 4  |-  ran  G  =  ran  G
43grpofo 20866 . . 3  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
5 fof 5451 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
6 fdm 5393 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
73grpon0 20869 . . . 4  |-  ( G  e.  GrpOp  ->  ran  G  =/=  (/) )
8 dmeq 4879 . . . . . 6  |-  ( dom 
G  =  ( ran 
G  X.  ran  G
)  ->  dom  dom  G  =  dom  ( ran  G  X.  ran  G ) )
9 dmxpid 4898 . . . . . 6  |-  dom  ( ran  G  X.  ran  G
)  =  ran  G
108, 9syl6eq 2331 . . . . 5  |-  ( dom 
G  =  ( ran 
G  X.  ran  G
)  ->  dom  dom  G  =  ran  G )
11 rneq 4904 . . . . . . 7  |-  ( dom 
G  =  ( ran 
G  X.  ran  G
)  ->  ran  dom  G  =  ran  ( ran  G  X.  ran  G ) )
12 rnxp 5106 . . . . . . . 8  |-  ( ran 
G  =/=  (/)  ->  ran  ( ran  G  X.  ran  G )  =  ran  G
)
13 eqtr 2300 . . . . . . . . 9  |-  ( ( ran  dom  G  =  ran  ( ran  G  X.  ran  G )  /\  ran  ( ran  G  X.  ran  G )  =  ran  G
)  ->  ran  dom  G  =  ran  G )
1413ex 423 . . . . . . . 8  |-  ( ran 
dom  G  =  ran  ( ran  G  X.  ran  G )  ->  ( ran  ( ran  G  X.  ran  G )  =  ran  G  ->  ran  dom  G  =  ran  G ) )
1512, 14syl5com 26 . . . . . . 7  |-  ( ran 
G  =/=  (/)  ->  ( ran  dom  G  =  ran  ( ran  G  X.  ran  G )  ->  ran  dom  G  =  ran  G ) )
16 eqtr 2300 . . . . . . . . 9  |-  ( ( dom  dom  G  =  ran  G  /\  ran  G  =  ran  dom  G )  ->  dom  dom  G  =  ran  dom  G )
1716expcom 424 . . . . . . . 8  |-  ( ran 
G  =  ran  dom  G  ->  ( dom  dom  G  =  ran  G  ->  dom  dom  G  =  ran  dom 
G ) )
1817eqcoms 2286 . . . . . . 7  |-  ( ran 
dom  G  =  ran  G  ->  ( dom  dom  G  =  ran  G  ->  dom  dom  G  =  ran  dom 
G ) )
1911, 15, 18syl56 30 . . . . . 6  |-  ( ran 
G  =/=  (/)  ->  ( dom  G  =  ( ran 
G  X.  ran  G
)  ->  ( dom  dom 
G  =  ran  G  ->  dom  dom  G  =  ran  dom  G ) ) )
2019com13 74 . . . . 5  |-  ( dom 
dom  G  =  ran  G  ->  ( dom  G  =  ( ran  G  X.  ran  G )  -> 
( ran  G  =/=  (/) 
->  dom  dom  G  =  ran  dom  G ) ) )
2110, 20mpcom 32 . . . 4  |-  ( dom 
G  =  ( ran 
G  X.  ran  G
)  ->  ( ran  G  =/=  (/)  ->  dom  dom  G  =  ran  dom  G )
)
226, 7, 21syl2im 34 . . 3  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  ( G  e.  GrpOp  ->  dom  dom  G  =  ran  dom  G )
)
234, 5, 223syl 18 . 2  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp  ->  dom  dom  G  =  ran  dom  G )
)
242, 2, 23sylc 56 1  |-  ( R  e.  RingOps  ->  dom  dom  G  =  ran  dom  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   GrpOpcgr 20853   RingOpscrngo 21042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-grpo 20858  df-ablo 20949  df-rngo 21043
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