Theorem rngorn1 37895

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 37872	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		grporndm 30544	. . 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 37894	. 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
7	4, 6	eqtrd 2780	1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  dom cdm 5700  ran crn 5701  ‘cfv 6575  1^st c1st 8030  2^nd c2nd 8031  GrpOpcgr 30523  RingOpscrngo 37856

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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772

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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fo 6581  df-fv 6583  df-ov 7453  df-1st 8032  df-2nd 8033  df-grpo 30527  df-ablo 30579  df-rngo 37857

This theorem is referenced by:  rngomndo  37897