Theorem rngorn1 37842

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 37819	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		grporndm 30533	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)
4	2, 3	syl 17	. 2 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐺)
5		rnplrnml0.1	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
6	5, 1	rngodm1dm2 37841	. 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
7	4, 6	eqtrd 2774	1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2103  dom cdm 5699  ran crn 5700  ‘cfv 6572  1^st c1st 8024  2^nd c2nd 8025  GrpOpcgr 30512  RingOpscrngo 37803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fo 6578  df-fv 6580  df-ov 7448  df-1st 8026  df-2nd 8027  df-grpo 30516  df-ablo 30568  df-rngo 37804

This theorem is referenced by:  rngomndo  37844