Theorem rngocom 38848

Description: Obsolete theorem, use ringcom 20509 instead. The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ringgcl.1	⊢ 𝐺 = (1st ‘𝑅)
ringgcl.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngocom	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Proof of Theorem rngocom

Step	Hyp	Ref	Expression
1		ringgcl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngoablo 38843	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ringgcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4	3	ablocom 31150	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
5	2, 4	syl3an1 1181	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ran crn 5651  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  AbelOpcablo 31146  RingOpscrngo 38829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-1st 8001  df-2nd 8002  df-ablo 31147  df-rngo 38830

This theorem is used by: (None)