Theorem isgrpNEW 17104

Description: The predicate "is a group."

Hypotheses

Ref	Expression
isgrp.0NEW	|- S = Struct(2, f, T. )
isgrp.1NEW	|- B = (base` G)
isgrp.2NEW	|- P = (+g` G)

Assertion

Ref	Expression
isgrpNEW	|- (G e. GrpNEW <-> (G e. S /\ A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e)))

Distinct variable groups:   a,b,c,e,f,m,B   f,G   P,a,b,c,e,f,m

Proof of Theorem isgrpNEW

Step	Hyp	Ref	Expression
1		df-grpNEW 17089	. . 3 |- GrpNEW = Struct(2, f, E.gE.p(g = (base` f) /\ p = (+g` f) /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e))))
2	1	eleq2i 1961	. 2 |- (G e. GrpNEW <-> G e. Struct(2, f, E.gE.p(g = (base` f) /\ p = (+g` f) /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)))))
3		isgrp.0NEW	. . 3 |- S = Struct(2, f, T. )
4		fveq2 4681	. . . . . . 7 |- (f = G -> (base` f) = (base` G))
5		isgrp.1NEW	. . . . . . 7 |- B = (base` G)
6	4, 5	syl6eqr 1946	. . . . . 6 |- (f = G -> (base` f) = B)
7	6	eqeq2d 1895	. . . . 5 |- (f = G -> (g = (base` f) <-> g = B))
8		fveq2 4681	. . . . . . 7 |- (f = G -> (+g` f) = (+g` G))
9		isgrp.2NEW	. . . . . . 7 |- P = (+g` G)
10	8, 9	syl6eqr 1946	. . . . . 6 |- (f = G -> (+g` f) = P)
11	10	eqeq2d 1895	. . . . 5 |- (f = G -> (p = (+g` f) <-> p = P))
12	7, 11	3anbi12d 1169	. . . 4 |- (f = G -> ((g = (base` f) /\ p = (+g` f) /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e))) <-> (g = B /\ p = P /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)))))
13	12	2exbidv 1659	. . 3 |- (f = G -> (E.gE.p(g = (base` f) /\ p = (+g` f) /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e))) <-> E.gE.p(g = B /\ p = P /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)))))
14	3, 13	elstr2 16718	. 2 |- (G e. Struct(2, f, E.gE.p(g = (base` f) /\ p = (+g` f) /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)))) <-> (G e. S /\ E.gE.p(g = B /\ p = P /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)))))
15		fvex 4689	. . . . . 6 |- (base` G) e. _V
16	5, 15	eqeltri 1967	. . . . 5 |- B e. _V
17		fvex 4689	. . . . . 6 |- (+g` G) e. _V
18	9, 17	eqeltri 1967	. . . . 5 |- P e. _V
19		eleq2 1958	. . . . . . . . . 10 |- (g = B -> ((apb) e. g <-> (apb) e. B))
20	19	anbi1d 679	. . . . . . . . 9 |- (g = B -> (((apb) e. g /\ ((apb)pc) = (ap(bpc))) <-> ((apb) e. B /\ ((apb)pc) = (ap(bpc)))))
21	20	raleqbi1dv 2271	. . . . . . . 8 |- (g = B -> (A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) <-> A.c e. B ((apb) e. B /\ ((apb)pc) = (ap(bpc)))))
22	21	raleqbi1dv 2271	. . . . . . 7 |- (g = B -> (A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) <-> A.b e. B A.c e. B ((apb) e. B /\ ((apb)pc) = (ap(bpc)))))
23	22	raleqbi1dv 2271	. . . . . 6 |- (g = B -> (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) <-> A.a e. B A.b e. B A.c e. B ((apb) e. B /\ ((apb)pc) = (ap(bpc)))))
24		rexeq 2267	. . . . . . . . 9 |- (g = B -> (E.m e. g (mpa) = e <-> E.m e. B (mpa) = e))
25	24	anbi2d 678	. . . . . . . 8 |- (g = B -> (((epa) = a /\ E.m e. g (mpa) = e) <-> ((epa) = a /\ E.m e. B (mpa) = e)))
26	25	raleqbi1dv 2271	. . . . . . 7 |- (g = B -> (A.a e. g ((epa) = a /\ E.m e. g (mpa) = e) <-> A.a e. B ((epa) = a /\ E.m e. B (mpa) = e)))
27	26	rexeqbi1dv 2272	. . . . . 6 |- (g = B -> (E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e) <-> E.e e. B A.a e. B ((epa) = a /\ E.m e. B (mpa) = e)))
28	23, 27	anbi12d 690	. . . . 5 |- (g = B -> ((A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)) <-> (A.a e. B A.b e. B A.c e. B ((apb) e. B /\ ((apb)pc) = (ap(bpc))) /\ E.e e. B A.a e. B ((epa) = a /\ E.m e. B (mpa) = e))))
29		opreq 4888	. . . . . . . . . 10 |- (p = P -> (apb) = (aPb))
30	29	eleq1d 1963	. . . . . . . . 9 |- (p = P -> ((apb) e. B <-> (aPb) e. B))
31		opreq 4888	. . . . . . . . . . 11 |- (p = P -> ((apb)pc) = ((apb)Pc))
32	29	opreq1d 4897	. . . . . . . . . . 11 |- (p = P -> ((apb)Pc) = ((aPb)Pc))
33	31, 32	eqtrd 1925	. . . . . . . . . 10 |- (p = P -> ((apb)pc) = ((aPb)Pc))
34		opreq 4888	. . . . . . . . . . 11 |- (p = P -> (ap(bpc)) = (aP(bpc)))
35		opreq 4888	. . . . . . . . . . . 12 |- (p = P -> (bpc) = (bPc))
36	35	opreq2d 4898	. . . . . . . . . . 11 |- (p = P -> (aP(bpc)) = (aP(bPc)))
37	34, 36	eqtrd 1925	. . . . . . . . . 10 |- (p = P -> (ap(bpc)) = (aP(bPc)))
38	33, 37	eqeq12d 1899	. . . . . . . . 9 |- (p = P -> (((apb)pc) = (ap(bpc)) <-> ((aPb)Pc) = (aP(bPc))))
39	30, 38	anbi12d 690	. . . . . . . 8 |- (p = P -> (((apb) e. B /\ ((apb)pc) = (ap(bpc))) <-> ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc)))))
40	39	ralbidv 2123	. . . . . . 7 |- (p = P -> (A.c e. B ((apb) e. B /\ ((apb)pc) = (ap(bpc))) <-> A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc)))))
41	40	2ralbidv 2140	. . . . . 6 |- (p = P -> (A.a e. B A.b e. B A.c e. B ((apb) e. B /\ ((apb)pc) = (ap(bpc))) <-> A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc)))))
42		opreq 4888	. . . . . . . . 9 |- (p = P -> (epa) = (ePa))
43	42	eqeq1d 1892	. . . . . . . 8 |- (p = P -> ((epa) = a <-> (ePa) = a))
44		opreq 4888	. . . . . . . . . 10 |- (p = P -> (mpa) = (mPa))
45	44	eqeq1d 1892	. . . . . . . . 9 |- (p = P -> ((mpa) = e <-> (mPa) = e))
46	45	rexbidv 2124	. . . . . . . 8 |- (p = P -> (E.m e. B (mpa) = e <-> E.m e. B (mPa) = e))
47	43, 46	anbi12d 690	. . . . . . 7 |- (p = P -> (((epa) = a /\ E.m e. B (mpa) = e) <-> ((ePa) = a /\ E.m e. B (mPa) = e)))
48	47	rexralbidv 2142	. . . . . 6 |- (p = P -> (E.e e. B A.a e. B ((epa) = a /\ E.m e. B (mpa) = e) <-> E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e)))
49	41, 48	anbi12d 690	. . . . 5 |- (p = P -> ((A.a e. B A.b e. B A.c e. B ((apb) e. B /\ ((apb)pc) = (ap(bpc))) /\ E.e e. B A.a e. B ((epa) = a /\ E.m e. B (mpa) = e)) <-> (A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e))))
50	16, 18, 28, 49	ceqsex2v 2328	. . . 4 |- (E.gE.p(g = B /\ p = P /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e))) <-> (A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e)))
51	50	anbi2i 538	. . 3 |- ((G e. S /\ E.gE.p(g = B /\ p = P /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)))) <-> (G e. S /\ (A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e))))
52		3anass 862	. . 3 |- ((G e. S /\ A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e)) <-> (G e. S /\ (A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e))))
53	51, 52	bitr4i 193	. 2 |- ((G e. S /\ E.gE.p(g = B /\ p = P /\ (A.a e. g A.b e. g A.c e. g ((apb) e. g /\ ((apb)pc) = (ap(bpc))) /\ E.e e. g A.a e. g ((epa) = a /\ E.m e. g (mpa) = e)))) <-> (G e. S /\ A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e)))
54	2, 14, 53	3bitri 194	1 |- (G e. GrpNEW <-> (G e. S /\ A.a e. B A.b e. B A.c e. B ((aPb) e. B /\ ((aPb)Pc) = (aP(bPc))) /\ E.e e. B A.a e. B ((ePa) = a /\ E.m e. B (mPa) = e)))

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   T. wtru 1260   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106  _Vcvv 2292  ` cfv 3998  (class class class)co 4884  2c2 7145  Structcstru 16707  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081

This theorem is referenced by:  grplem1 17105  grplidinvNEW 17108  isgrpiNEW 17115

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-tru 1262  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-fv 4014  df-opr 4886  df-struct 16708  df-grpNEW 17089

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