Theorem grppropstrg 13567

Description: Generalize a specific 2-element group L to show that any set K with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grppropstr.b	|- ( Base ` K ) = B
grppropstr.p	|- ( +g ` K ) = .+
grppropstr.l	|- L = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }

Assertion

Ref	Expression
grppropstrg	|- ( K e. V -> ( K e. Grp <-> L e. Grp ) )

Proof of Theorem grppropstrg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grppropstr.b	. . . . 5 |- ( Base ` K ) = B
2		basfn 13106	. . . . . 6 |- Base Fn _V
3		elex 2811	. . . . . 6 |- ( K e. V -> K e. _V )
4		funfvex 5646	. . . . . . 7 |- ( ( Fun Base /\ K e. dom Base ) -> ( Base ` K ) e. _V )
5	4	funfni 5423	. . . . . 6 |- ( ( Base Fn _V /\ K e. _V ) -> ( Base ` K ) e. _V )
6	2, 3, 5	sylancr 414	. . . . 5 |- ( K e. V -> ( Base ` K ) e. _V )
7	1, 6	eqeltrrid 2317	. . . 4 |- ( K e. V -> B e. _V )
8		grppropstr.p	. . . . 5 |- ( +g ` K ) = .+
9		plusgslid 13160	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
10	9	slotex 13074	. . . . 5 |- ( K e. V -> ( +g ` K ) e. _V )
11	8, 10	eqeltrrid 2317	. . . 4 |- ( K e. V -> .+ e. _V )
12		grppropstr.l	. . . . 5 |- L = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }
13	12	grpbaseg 13175	. . . 4 |- ( ( B e. _V /\ .+ e. _V ) -> B = ( Base ` L ) )
14	7, 11, 13	syl2anc 411	. . 3 |- ( K e. V -> B = ( Base ` L ) )
15	1, 14	eqtrid 2274	. . 3 |- ( K e. V -> ( Base ` K ) = ( Base ` L ) )
16	14, 15	eqtr4d 2265	. 2 |- ( K e. V -> B = ( Base ` K ) )
17	12	grpplusgg 13176	. . . . 5 |- ( ( B e. _V /\ .+ e. _V ) -> .+ = ( +g ` L ) )
18	7, 11, 17	syl2anc 411	. . . 4 |- ( K e. V -> .+ = ( +g ` L ) )
19	8, 18	eqtrid 2274	. . 3 |- ( K e. V -> ( +g ` K ) = ( +g ` L ) )
20	19	oveqdr 6035	. 2 |- ( ( K e. V /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
21	16, 14, 20	grppropd 13565	1 |- ( K e. V -> ( K e. Grp <-> L e. Grp ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   {cpr 3667   <.cop 3669   Fn wfn 5313   ` cfv 5318   ndxcnx 13044   Basecbs 13047   +g cplusg 13125   Grpcgrp 13548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-ov 6010  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-ndx 13050  df-slot 13051  df-base 13053  df-plusg 13138  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551

This theorem is referenced by:  ring1  14037