Theorem grpinv 8069

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		ssid 2080	. . . . 5 |- X (_ X
2		pm3.26 319	. . . . . . 7 |- (((yGA) = U /\ (AGy) = U) -> (yGA) = U)
3	2	a1i 8	. . . . . 6 |- (y e. X -> (((yGA) = U /\ (AGy) = U) -> (yGA) = U))
4	3	rgen 1698	. . . . 5 |- A.y e. X (((yGA) = U /\ (AGy) = U) -> (yGA) = U)
5		reuuniss2 2891	. . . . 5 |- (((X (_ 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})
6	1, 4, 5	mpanl12 708	. . . 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})
7		grpinv.1	. . . . . 6 |- X = ran G
8		grpinv.2	. . . . . 6 |- U = (Id` G)
9	7, 8	grpidinv2 8060	. . . . 5 |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
10	9	pm3.27d 325	. . . 4 |- ((G e. Grp /\ A e. X) -> E.y e. X ((yGA) = U /\ (AGy) = U))
11	7, 8	grpinveu 8064	. . . 4 |- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)
12	6, 10, 11	sylanc 471	. . 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	7, 8, 13	grpinvval 8067	. . 3 |- ((G e. Grp /\ A e. X) -> (N` A) = U.{y e. X | (yGA) = U})
15	12, 14	eqtr4d 1510	. 2 |- ((G e. Grp /\ A e. X) -> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = (N` A))
16		opreq1 3968	. . . . . 6 |- (y = (N` A) -> (yGA) = ((N` A)GA))
17	16	eqeq1d 1483	. . . . 5 |- (y = (N` A) -> ((yGA) = U <-> ((N` A)GA) = U))
18		opreq2 3969	. . . . . 6 |- (y = (N` A) -> (AGy) = (AG(N` A)))
19	18	eqeq1d 1483	. . . . 5 |- (y = (N` A) -> ((AGy) = U <-> (AG(N` A)) = U))
20	17, 19	anbi12d 628	. . . 4 |- (y = (N` A) -> (((yGA) = U /\ (AGy) = U) <-> (((N` A)GA) = U /\ (AG(N` A)) = U)))
21	20	reuuni2 2884	. . 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)))
22	7, 13	grpinvcl 8068	. . 3 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
23		reuss2 2275	. . . . 5 |- (((X (_ 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))
24	1, 4, 23	mpanl12 708	. . . 4 |- ((E.y e. X ((yGA) = U /\ (AGy) = U) /\ E!y e. X (yGA) = U) -> E!y e. X ((yGA) = U /\ (AGy) = U))
25	24, 10, 11	sylanc 471	. . 3 |- ((G e. Grp /\ A e. X) -> E!y e. X ((yGA) = U /\ (AGy) = U))
26	21, 22, 25	sylanc 471	. 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 196	1 |- ((G e. Grp /\ A e. X) -> (((N` A)GA) = U /\ (AG(N` A)) = U))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  E!wreu 1647  {crab 1648   (_ wss 2047  U.cuni 2503  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035

This theorem is referenced by:  grplinv 8070  grprinv 8071

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-rep 2693  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  df-ginv 8039

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