Theorem grpinv 8065

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 2083	. . . . 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 1701	. . . . 5 |- A.y e. X (((yGA) = U /\ (AGy) = U) -> (yGA) = U)
5		reuuniss2 2897	. . . . 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 710	. . . 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 8056	. . . . 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 8060	. . . 4 |- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)
12	6, 10, 11	sylanc 473	. . 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 8063	. . 3 |- ((G e. Grp /\ A e. X) -> (N` A) = U.{y e. X | (yGA) = U})
15	12, 14	eqtr4d 1513	. 2 |- ((G e. Grp /\ A e. X) -> U.{y e. X | ((yGA) = U /\ (AGy) = U)} = (N` A))
16		opreq1 3974	. . . . . 6 |- (y = (N` A) -> (yGA) = ((N` A)GA))
17	16	eqeq1d 1486	. . . . 5 |- (y = (N` A) -> ((yGA) = U <-> ((N` A)GA) = U))
18		opreq2 3975	. . . . . 6 |- (y = (N` A) -> (AGy) = (AG(N` A)))
19	18	eqeq1d 1486	. . . . 5 |- (y = (N` A) -> ((AGy) = U <-> (AG(N` A)) = U))
20	17, 19	anbi12d 630	. . . 4 |- (y = (N` A) -> (((yGA) = U /\ (AGy) = U) <-> (((N` A)GA) = U /\ (AG(N` A)) = U)))
21	20	reuuni2 2890	. . 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 8064	. . 3 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
23		reuss2 2278	. . . . 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 710	. . . 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 473	. . 3 |- ((G e. Grp /\ A e. X) -> E!y e. X ((yGA) = U /\ (AGy) = U))
26	21, 22, 25	sylanc 473	. 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 958   e. wcel 960  A.wral 1648  E.wrex 1649  E!wreu 1650  {crab 1651   (_ wss 2050  U.cuni 2507  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031  invcgn 8032

This theorem is referenced by:  grplinv 8066  grprinv 8067

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-rep 2698  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  df-ginv 8036

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