Theorem ringabl 8234

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 ∈ Ring → G ∈ Abel)

Proof of Theorem ringabl

Step	Hyp	Ref	Expression
1		ringabl.1	. . . 4 ⊢ G = (1st ‘R)
2		eqid 1522	. . . 4 ⊢ (2nd ‘R) = (2nd ‘R)
3		eqid 1522	. . . 4 ⊢ ran G = ran G
4	1, 2, 3	ringi 8226	. . 3 ⊢ (R ∈ Ring → ((G ∈ Abel ⋀ (2nd ‘R):(ran G × ran G)–→ran G) ⋀ (∀x ∈ ran G∀y ∈ ran G∀z ∈ 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))) ⋀ ∃x ∈ ran G∀y ∈ ran G((y(2nd ‘R)x) = y ⋀ (x(2nd ‘R)y) = y))))
5	4	pm3.26d 328	. 2 ⊢ (R ∈ Ring → (G ∈ Abel ⋀ (2nd ‘R):(ran G × ran G)–→ran G))
6	5	pm3.26d 328	1 ⊢ (R ∈ Ring → G ∈ Abel)

Syntax hints:   → wi 3   ⋀ wa 230   ⋀ w3a 787   = wceq 997   ∈ wcel 999  ∀wral 1692  ∃wrex 1693   × cxp 3225  ran crn 3228  –→wf 3235   ‘cfv 3239  (class class class)co 4021  1^st c1st 4135  2^nd c2nd 4136  Abelcabl 8183  Ringcring 8223

This theorem is referenced by:  ringgrp 8235  ringcom 8237  ringa23 8239  ringa4 8240

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-fv 3255  df-opr 4023  df-1st 4137  df-2nd 4138  df-ring 8224

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