Theorem rngorn1 38612

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 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Proof of Theorem rngorn1

Step	Hyp	Ref	Expression
1		rnplrnml0.2	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 38589	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		grporndm 30873	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)
4	2, 3	syl 18	. 2 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐺)
5		rnplrnml0.1	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
6	5, 1	rngodm1dm2 38611	. 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
7	4, 6	eqtrd 2797	1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  dom cdm 5660  ran crn 5661  ‘cfv 6536  1^st c1st 7982  2^nd c2nd 7983  GrpOpcgr 30852  RingOpscrngo 38573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7415  df-1st 7984  df-2nd 7985  df-grpo 30856  df-ablo 30908  df-rngo 38574

This theorem is used by:  rngomndo  38614