Theorem rngorn1 22012

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 21983	. . 3 |- ( R e. RingOps -> G e. GrpOp )
3		grporndm 21803	. . 3 |- ( G e. GrpOp -> ran G = dom dom G )
4	2, 3	syl 16	. 2 |- ( R e. RingOps -> ran G = dom dom G )
5		rnplrnml0.1	. . 3 |- H = ( 2nd ` R )
6	5, 1	rngodm1dm2 22011	. 2 |- ( R e. RingOps -> dom dom G = dom dom H )
7	4, 6	eqtrd 2470	1 |- ( R e. RingOps -> ran G = dom dom H )

Syntax hints:   -> wi 4   = wceq 1653   e. wcel 1726   dom cdm 4881   ran crn 4882   ` cfv 5457   1stc1st 6350   2ndc2nd 6351   GrpOpcgr 21779   RingOpscrngo 21968

This theorem is referenced by:  rngomndo  22014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-grpo 21784  df-ablo 21875  df-rngo 21969