Theorem rngorn1 21964

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 21935	. . 3 |- ( R e. RingOps -> G e. GrpOp )
3		grporndm 21755	. . 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 21963	. 2 |- ( R e. RingOps -> dom dom G = dom dom H )
7	4, 6	eqtrd 2440	1 |- ( R e. RingOps -> ran G = dom dom H )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   dom cdm 4841   ran crn 4842   ` cfv 5417   1stc1st 6310   2ndc2nd 6311   GrpOpcgr 21731   RingOpscrngo 21920

This theorem is referenced by:  rngomndo  21966

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-ov 6047  df-1st 6312  df-2nd 6313  df-grpo 21736  df-ablo 21827  df-rngo 21921