Theorem grpinv 9353

Description: The properties of a group element's inverse.

Hypotheses

Ref	Expression
grpinv.1	|- X = ran G
grpinv.2	|- U = (Id` G)
grpinv.3	|- N = (inv` G)

Assertion

Ref	Expression
grpinv	|- ((G e. Grp /\ A e. X) -> (((N` A)GA) = U /\ (AG(N` A)) = U))

Proof of Theorem grpinv

Step	Hyp	Ref	Expression
1		grpinv.1	. . . . . 6 |- X = ran G
2		grpinv.2	. . . . . 6 |- U = (Id` G)
3	1, 2	grpidinv2 9344	. . . . 5 |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
4	3	simprd 352	. . . 4 |- ((G e. Grp /\ A e. X) -> E.y e. X ((yGA) = U /\ (AGy) = U))
5	1, 2	grpinveu 9348	. . . 4 |- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)
6		ssid 2634	. . . . 5 |- X C_ X
7		simpl 346	. . . . . . 7 |- (((yGA) = U /\ (AGy) = U) -> (yGA) = U)
8	7	a1i 8	. . . . . 6 |- (y e. X -> (((yGA) = U /\ (AGy) = U) -> (yGA) = U))
9	8	rgen 2159	. . . . 5 |- A.y e. X (((yGA) = U /\ (AGy) = U) -> (yGA) = U)
10		reuuniss2 3817	. . . . 5 |- (((X C_ X /\ A.y e. X (((yGA) = U /\ (AGy) = U) -> (yGA) = U)) /\ (E.y e. X ((yGA) = U /\ (AGy) = U) /\ E!y e. X (yGA) = U)) -> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = U.{y e. X | (yGA) = U})
11	6, 9, 10	mpanl12 773	. . . 4 |- ((E.y e. X ((yGA) = U /\ (AGy) = U) /\ E!y e. X (yGA) = U) -> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = U.{y e. X | (yGA) = U})
12	4, 5, 11	syl11anc 524	. . 3 |- ((G e. Grp /\ A e. X) -> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = U.{y e. X | (yGA) = U})
13		grpinv.3	. . . 4 |- N = (inv` G)
14	1, 2, 13	grpinvval 9351	. . 3 |- ((G e. Grp /\ A e. X) -> (N` A) = U.{y e. X | (yGA) = U})
15	12, 14	eqtr4d 1928	. 2 |- ((G e. Grp /\ A e. X) -> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = (N` A))
16	1, 13	grpinvcl 9352	. . 3 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
17		reuss2 2870	. . . . 5 |- (((X C_ X /\ A.y e. X (((yGA) = U /\ (AGy) = U) -> (yGA) = U)) /\ (E.y e. X ((yGA) = U /\ (AGy) = U) /\ E!y e. X (yGA) = U)) -> E!y e. X ((yGA) = U /\ (AGy) = U))
18	6, 9, 17	mpanl12 773	. . . 4 |- ((E.y e. X ((yGA) = U /\ (AGy) = U) /\ E!y e. X (yGA) = U) -> E!y e. X ((yGA) = U /\ (AGy) = U))
19	4, 5, 18	syl11anc 524	. . 3 |- ((G e. Grp /\ A e. X) -> E!y e. X ((yGA) = U /\ (AGy) = U))
20		opreq1 4889	. . . . . 6 |- (y = (N` A) -> (yGA) = ((N` A)GA))
21	20	eqeq1d 1892	. . . . 5 |- (y = (N` A) -> ((yGA) = U <-> ((N` A)GA) = U))
22		opreq2 4890	. . . . . 6 |- (y = (N` A) -> (AGy) = (AG(N` A)))
23	22	eqeq1d 1892	. . . . 5 |- (y = (N` A) -> ((AGy) = U <-> (AG(N` A)) = U))
24	21, 23	anbi12d 690	. . . 4 |- (y = (N` A) -> (((yGA) = U /\ (AGy) = U) <-> (((N` A)GA) = U /\ (AG(N` A)) = U)))
25	24	reuuni2 3811	. . 3 |- (((N` A) e. X /\ E!y e. X ((yGA) = U /\ (AGy) = U)) -> ((((N` A)GA) = U /\ (AG(N` A)) = U) <-> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = (N` A)))
26	16, 19, 25	syl11anc 524	. 2 |- ((G e. Grp /\ A e. X) -> ((((N` A)GA) = U /\ (AG(N` A)) = U) <-> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = (N` A)))
27	15, 26	mpbird 213	1 |- ((G e. Grp /\ A e. X) -> (((N` A)GA) = U /\ (AG(N` A)) = U))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108   C_ wss 2593  U.cuni 3177  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  invcgn 9313

This theorem is referenced by:  grplinv 9354  grprinv 9355

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316  df-gid 9317  df-ginv 9318

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