Theorem isgrpoi 27352

Description: Properties that determine a group operation. Read N as N ( x ). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isgrpoi.1	|- X e. _V
isgrpoi.2	|- G : ( X X. X ) --> X
isgrpoi.3	|- ( ( x e. X /\ y e. X /\ z e. X ) -> ( ( x G y ) G z ) = ( x G ( y G z ) ) )
isgrpoi.4	|- U e. X
isgrpoi.5	|- ( x e. X -> ( U G x ) = x )
isgrpoi.6	|- ( x e. X -> N e. X )
isgrpoi.7	|- ( x e. X -> ( N G x ) = U )

Assertion

Ref	Expression
isgrpoi	|- G e. GrpOp

Distinct variable groups:   x, y, z, G   x, U, y, z   x, X, y, z   y, N

Allowed substitution hints:   N( x, z)

Proof of Theorem isgrpoi

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgrpoi.2	. 2 |- G : ( X X. X ) --> X
2		isgrpoi.3	. . 3 |- ( ( x e. X /\ y e. X /\ z e. X ) -> ( ( x G y ) G z ) = ( x G ( y G z ) ) )
3	2	rgen3 2976	. 2 |- A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) )
4		isgrpoi.4	. . 3 |- U e. X
5		isgrpoi.5	. . . . 5 |- ( x e. X -> ( U G x ) = x )
6		isgrpoi.6	. . . . . 6 |- ( x e. X -> N e. X )
7		isgrpoi.7	. . . . . 6 |- ( x e. X -> ( N G x ) = U )
8		oveq1 6657	. . . . . . . 8 |- ( y = N -> ( y G x ) = ( N G x ) )
9	8	eqeq1d 2624	. . . . . . 7 |- ( y = N -> ( ( y G x ) = U <-> ( N G x ) = U ) )
10	9	rspcev 3309	. . . . . 6 |- ( ( N e. X /\ ( N G x ) = U ) -> E. y e. X ( y G x ) = U )
11	6, 7, 10	syl2anc 693	. . . . 5 |- ( x e. X -> E. y e. X ( y G x ) = U )
12	5, 11	jca 554	. . . 4 |- ( x e. X -> ( ( U G x ) = x /\ E. y e. X ( y G x ) = U ) )
13	12	rgen 2922	. . 3 |- A. x e. X ( ( U G x ) = x /\ E. y e. X ( y G x ) = U )
14		oveq1 6657	. . . . . . 7 |- ( u = U -> ( u G x ) = ( U G x ) )
15	14	eqeq1d 2624	. . . . . 6 |- ( u = U -> ( ( u G x ) = x <-> ( U G x ) = x ) )
16		eqeq2 2633	. . . . . . 7 |- ( u = U -> ( ( y G x ) = u <-> ( y G x ) = U ) )
17	16	rexbidv 3052	. . . . . 6 |- ( u = U -> ( E. y e. X ( y G x ) = u <-> E. y e. X ( y G x ) = U ) )
18	15, 17	anbi12d 747	. . . . 5 |- ( u = U -> ( ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) <-> ( ( U G x ) = x /\ E. y e. X ( y G x ) = U ) ) )
19	18	ralbidv 2986	. . . 4 |- ( u = U -> ( A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) <-> A. x e. X ( ( U G x ) = x /\ E. y e. X ( y G x ) = U ) ) )
20	19	rspcev 3309	. . 3 |- ( ( U e. X /\ A. x e. X ( ( U G x ) = x /\ E. y e. X ( y G x ) = U ) ) -> E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) )
21	4, 13, 20	mp2an 708	. 2 |- E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u )
22		isgrpoi.1	. . . . 5 |- X e. _V
23	22, 22	xpex 6962	. . . 4 |- ( X X. X ) e. _V
24		fex 6490	. . . 4 |- ( ( G : ( X X. X ) --> X /\ ( X X. X ) e. _V ) -> G e. _V )
25	1, 23, 24	mp2an 708	. . 3 |- G e. _V
26	5	eqcomd 2628	. . . . . . . . 9 |- ( x e. X -> x = ( U G x ) )
27		rspceov 6692	. . . . . . . . . 10 |- ( ( U e. X /\ x e. X /\ x = ( U G x ) ) -> E. y e. X E. z e. X x = ( y G z ) )
28	4, 27	mp3an1 1411	. . . . . . . . 9 |- ( ( x e. X /\ x = ( U G x ) ) -> E. y e. X E. z e. X x = ( y G z ) )
29	26, 28	mpdan 702	. . . . . . . 8 |- ( x e. X -> E. y e. X E. z e. X x = ( y G z ) )
30	29	rgen 2922	. . . . . . 7 |- A. x e. X E. y e. X E. z e. X x = ( y G z )
31		foov 6808	. . . . . . 7 |- ( G : ( X X. X ) -onto-> X <-> ( G : ( X X. X ) --> X /\ A. x e. X E. y e. X E. z e. X x = ( y G z ) ) )
32	1, 30, 31	mpbir2an 955	. . . . . 6 |- G : ( X X. X ) -onto-> X
33		forn 6118	. . . . . 6 |- ( G : ( X X. X ) -onto-> X -> ran G = X )
34	32, 33	ax-mp 5	. . . . 5 |- ran G = X
35	34	eqcomi 2631	. . . 4 |- X = ran G
36	35	isgrpo 27351	. . 3 |- ( G e. _V -> ( G e. GrpOp <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) ) ) )
37	25, 36	ax-mp 5	. 2 |- ( G e. GrpOp <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) ) )
38	1, 3, 21, 37	mpbir3an 1244	1 |- G e. GrpOp

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   X. cxp 5112   ran crn 5115   -->wf 5884   -onto->wfo 5886  (class class class)co 6650   GrpOpcgr 27343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-grpo 27347

This theorem is referenced by:  cnaddabloOLD  27436  hilablo  28017  hhssabloilem  28118  grposnOLD  33681