Theorem grpid 8061

Description: Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

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

Assertion

Ref	Expression
grpid	|- ((G e. Grp /\ A e. X) -> (A = U <-> (AGA) = A))

Proof of Theorem grpid

Step	Hyp	Ref	Expression
1		grpinveu.1	. . . . . 6 |- X = ran G
2		grpinveu.2	. . . . . 6 |- U = (Id` G)
3	1, 2	grpidcl 8055	. . . . 5 |- (G e. Grp -> U e. X)
4	1	grprcan 8059	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ U e. X /\ A e. X)) -> ((AGA) = (UGA) <-> A = U))
5	4	3exp2 853	. . . . 5 |- (G e. Grp -> (A e. X -> (U e. X -> (A e. X -> ((AGA) = (UGA) <-> A = U)))))
6	3, 5	mpid 47	. . . 4 |- (G e. Grp -> (A e. X -> (A e. X -> ((AGA) = (UGA) <-> A = U))))
7	6	pm2.43d 65	. . 3 |- (G e. Grp -> (A e. X -> ((AGA) = (UGA) <-> A = U)))
8	7	imp 350	. 2 |- ((G e. Grp /\ A e. X) -> ((AGA) = (UGA) <-> A = U))
9	1, 2	grplid 8057	. . 3 |- ((G e. Grp /\ A e. X) -> (UGA) = A)
10	9	eqeq2d 1489	. 2 |- ((G e. Grp /\ A e. X) -> ((AGA) = (UGA) <-> (AGA) = A))
11	8, 10	bitr3d 532	1 |- ((G e. Grp /\ A e. X) -> (A = U <-> (AGA) = A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031

This theorem is referenced by:  subgid 8116  hhssnv 9129  ghomgrpilem2 10381  ghomid 10389  symgidi 10402

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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