Theorem grpinveu 8064

Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55.

Hypotheses

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

Assertion

Ref	Expression
grpinveu	|- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)

Distinct variable groups:   y,A   y,G   y,U   y,X

Proof of Theorem grpinveu

Step	Hyp	Ref	Expression
1		grpinveu.1	. . . . 5 |- X = ran G
2		grpinveu.2	. . . . 5 |- U = (Id` G)
3	1, 2	grpidinv2 8060	. . . 4 |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
4		pm3.26 319	. . . . . 6 |- (((yGA) = U /\ (AGy) = U) -> (yGA) = U)
5	4	r19.22si 1734	. . . . 5 |- (E.y e. X ((yGA) = U /\ (AGy) = U) -> E.y e. X (yGA) = U)
6	5	adantl 388	. . . 4 |- ((((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)) -> E.y e. X (yGA) = U)
7	3, 6	syl 10	. . 3 |- ((G e. Grp /\ A e. X) -> E.y e. X (yGA) = U)
8	1	grprcan 8063	. . . . . . . . . . . 12 |- ((G e. Grp /\ (y e. X /\ z e. X /\ A e. X)) -> ((yGA) = (zGA) <-> y = z))
9		eqtr3t 1494	. . . . . . . . . . . 12 |- (((yGA) = U /\ (zGA) = U) -> (yGA) = (zGA))
10	8, 9	syl5bi 208	. . . . . . . . . . 11 |- ((G e. Grp /\ (y e. X /\ z e. X /\ A e. X)) -> (((yGA) = U /\ (zGA) = U) -> y = z))
11	10	3exp2 851	. . . . . . . . . 10 |- (G e. Grp -> (y e. X -> (z e. X -> (A e. X -> (((yGA) = U /\ (zGA) = U) -> y = z)))))
12	11	com24 37	. . . . . . . . 9 |- (G e. Grp -> (A e. X -> (z e. X -> (y e. X -> (((yGA) = U /\ (zGA) = U) -> y = z)))))
13	12	imp41 368	. . . . . . . 8 |- ((((G e. Grp /\ A e. X) /\ z e. X) /\ y e. X) -> (((yGA) = U /\ (zGA) = U) -> y = z))
14	13	an1rs 489	. . . . . . 7 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ z e. X) -> (((yGA) = U /\ (zGA) = U) -> y = z))
15	14	exp3a 375	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ z e. X) -> ((yGA) = U -> ((zGA) = U -> y = z)))
16	15	r19.21adva 1719	. . . . 5 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((yGA) = U -> A.z e. X ((zGA) = U -> y = z)))
17	16	ancld 298	. . . 4 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((yGA) = U -> ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z))))
18	17	r19.22dva 1739	. . 3 |- ((G e. Grp /\ A e. X) -> (E.y e. X (yGA) = U -> E.y e. X ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z))))
19	7, 18	mpd 26	. 2 |- ((G e. Grp /\ A e. X) -> E.y e. X ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z)))
20		opreq1 3968	. . . 4 |- (y = z -> (yGA) = (zGA))
21	20	eqeq1d 1483	. . 3 |- (y = z -> ((yGA) = U <-> (zGA) = U))
22	21	reu8 1936	. 2 |- (E!y e. X (yGA) = U <-> E.y e. X ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z)))
23	19, 22	sylibr 200	1 |- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  E!wreu 1647  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034

This theorem is referenced by:  grpinvcl 8068  grpinv 8069

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037  df-gid 8038

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