Theorem rngorn1 38274

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 38251	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		grporndm 30600	. . 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 38273	. 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
7	4, 6	eqtrd 2772	1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  dom cdm 5626  ran crn 5627  ‘cfv 6494  1^st c1st 7935  2^nd c2nd 7936  GrpOpcgr 30579  RingOpscrngo 38235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-fo 6500  df-fv 6502  df-ov 7365  df-1st 7937  df-2nd 7938  df-grpo 30583  df-ablo 30635  df-rngo 38236

This theorem is referenced by:  rngomndo  38276