Theorem ringabl 8146

Description: A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Hypothesis

Ref	Expression
ringabl.1	|- G = (1st` R)

Assertion

Ref	Expression
ringabl	|- (R e. Ring -> G e. Abel)

Proof of Theorem ringabl

Step	Hyp	Ref	Expression
1		ringabl.1	. . . 4 |- G = (1st` R)
2		eqid 1478	. . . 4 |- (2nd` R) = (2nd` R)
3		eqid 1478	. . . 4 |- ran G = ran G
4	1, 2, 3	ringi 8138	. . 3 |- (R e. Ring -> ((G e. Abel /\ (2nd` R):(ran G X. ran G)-->ran G) /\ (A.x e. ran GA.y e. ran GA.z e. ran G(((x(2nd` R)y)(2nd` R)z) = (x(2nd` R)(y(2nd` R)z)) /\ (x(2nd` R)(yGz)) = ((x(2nd` R)y)G(x(2nd` R)z)) /\ ((xGy)(2nd` R)z) = ((x(2nd` R)z)G(y(2nd` R)z))) /\ E.x e. ran GA.y e. ran G((y(2nd` R)x) = y /\ (x(2nd` R)y) = y))))
5	4	pm3.26d 321	. 2 |- (R e. Ring -> (G e. Abel /\ (2nd` R):(ran G X. ran G)-->ran G))
6	5	pm3.26d 321	1 |- (R e. Ring -> G e. Abel)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   X. cxp 3174  ran crn 3177  -->wf 3184  ` cfv 3188  (class class class)co 3969  1stc1st 4083  2ndc2nd 4084  Abelcabl 8095  Ringcring 8135

This theorem is referenced by:  ringgrp 8147  ringcom 8149  ringa23 8151  ringa4 8152

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-ring 8136

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