Theorem rngorn1 21086

Description: In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rnplrnml0.1	|- H = ( 2nd ` R )
rnplrnml0.2	|- G = ( 1st ` R )

Assertion

Ref	Expression
rngorn1	|- ( R e. RingOps -> ran G = dom dom H )

Proof of Theorem rngorn1

Step	Hyp	Ref	Expression
1		rnplrnml0.2	. . . 4 |- G = ( 1st ` R )
2	1	rngogrpo 21057	. . 3 |- ( R e. RingOps -> G e. GrpOp )
3		grporndm 20877	. . 3 |- ( G e. GrpOp -> ran G = dom dom G )
4	2, 3	syl 15	. 2 |- ( R e. RingOps -> ran G = dom dom G )
5		rnplrnml0.1	. . 3 |- H = ( 2nd ` R )
6	5, 1	rngodm1dm2 21085	. 2 |- ( R e. RingOps -> dom dom G = dom dom H )
7	4, 6	eqtrd 2315	1 |- ( R e. RingOps -> ran G = dom dom H )

Syntax hints:   -> wi 4   = wceq 1623   e. wcel 1684   dom cdm 4689   ran crn 4690   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853   RingOpscrngo 21042

This theorem is referenced by:  rngomndo  21088  rngodmdmrn  25418

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