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Theorem rngodmeqrn 25522
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 21073 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 eqid 2296 . . . 4  |-  ran  G  =  ran  G
43grpofo 20882 . . 3  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
5 fof 5467 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
6 fdm 5409 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
73grpon0 20885 . . . 4  |-  ( G  e.  GrpOp  ->  ran  G  =/=  (/) )
8 dmeq 4895 . . . . . 6  |-  ( dom 
G  =  ( ran 
G  X.  ran  G
)  ->  dom  dom  G  =  dom  ( ran  G  X.  ran  G ) )
9 dmxpid 4914 . . . . . 6  |-  dom  ( ran  G  X.  ran  G
)  =  ran  G
108, 9syl6eq 2344 . . . . 5  |-  ( dom 
G  =  ( ran 
G  X.  ran  G
)  ->  dom  dom  G  =  ran  G )
11 rneq 4920 . . . . . . 7  |-  ( dom 
G  =  ( ran 
G  X.  ran  G
)  ->  ran  dom  G  =  ran  ( ran  G  X.  ran  G ) )
12 rnxp 5122 . . . . . . . 8  |-  ( ran 
G  =/=  (/)  ->  ran  ( ran  G  X.  ran  G )  =  ran  G
)
13 eqtr 2313 . . . . . . . . 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 2313 . . . . . . . . 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 2299 . . . . . . 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 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468    X. cxp 4703   dom cdm 4705   ran crn 4706   -->wf 5267   -onto->wfo 5269   ` cfv 5271   1stc1st 6136   GrpOpcgr 20869   RingOpscrngo 21058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-grpo 20874  df-ablo 20965  df-rngo 21059
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