Theorem grprid 8058

Description: The identity element of a group is a right identity.

Hypotheses

Ref	Expression
grpidval.1	|- X = ran G
grpidval.2	|- U = (Id` G)

Assertion

Ref	Expression
grprid	|- ((G e. Grp /\ A e. X) -> (AGU) = A)

Proof of Theorem grprid

Step	Hyp	Ref	Expression
1		grpidval.1	. . 3 |- X = ran G
2		grpidval.2	. . 3 |- U = (Id` G)
3	1, 2	grpidinv2 8056	. 2 |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
4		simplr 415	. 2 |- ((((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)) -> (AGU) = A)
5	3, 4	syl 10	1 |- ((G e. Grp /\ A e. X) -> (AGU) = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wrex 1649  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031

This theorem is referenced by:  grprcan 8059  grpinvid1 8068  grpinvid2 8069  grppncan 8086  grpnpcan 8087  ring0rid 8156  vc0rid 8182  vcm 8186  nv0rid 8252  cayleylem3 10406

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-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-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  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-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034  df-gid 8035

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