Theorem rngorn1 37312

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 37289	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		grporndm 30268	. . 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 37311	. 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
7	4, 6	eqtrd 2766	1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  dom cdm 5669  ran crn 5670  ‘cfv 6536  1^st c1st 7969  2^nd c2nd 7970  GrpOpcgr 30247  RingOpscrngo 37273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7407  df-1st 7971  df-2nd 7972  df-grpo 30251  df-ablo 30303  df-rngo 37274

This theorem is referenced by:  rngomndo  37314