Theorem tgrpset 36033

Description: The translation group for a fiducial co-atom W. (Contributed by NM, 5-Jun-2013.)

Hypotheses

Ref	Expression
tgrpset.h	|- H = ( LHyp ` K )
tgrpset.t	|- T = ( ( LTrn ` K ) ` W )
tgrpset.g	|- G = ( ( TGrp ` K ) ` W )

Assertion

Ref	Expression
tgrpset	|- ( ( K e. V /\ W e. H ) -> G = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } )

Distinct variable groups:   f, g, K   T, f, g   f, W, g

Allowed substitution hints:   G( f, g)   H( f, g)   V( f, g)

Proof of Theorem tgrpset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgrpset.g	. 2 |- G = ( ( TGrp ` K ) ` W )
2		tgrpset.h	. . . . 5 |- H = ( LHyp ` K )
3	2	tgrpfset 36032	. . . 4 |- ( K e. V -> ( TGrp ` K ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) )
4	3	fveq1d 6193	. . 3 |- ( K e. V -> ( ( TGrp ` K ) ` W ) = ( ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) ` W ) )
5		fveq2 6191	. . . . . . 7 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
6	5	opeq2d 4409	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. = <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. )
7		eqidd 2623	. . . . . . . 8 |- ( w = W -> ( f o. g ) = ( f o. g ) )
8	5, 5, 7	mpt2eq123dv 6717	. . . . . . 7 |- ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) = ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) )
9	8	opeq2d 4409	. . . . . 6 |- ( w = W -> <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. = <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. )
10	6, 9	preq12d 4276	. . . . 5 |- ( w = W -> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } = { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. } )
11		eqid 2622	. . . . 5 |- ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } )
12		prex 4909	. . . . 5 |- { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. } e. _V
13	10, 11, 12	fvmpt 6282	. . . 4 |- ( W e. H -> ( ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) ` W ) = { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. } )
14		tgrpset.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
15	14	opeq2i 4406	. . . . 5 |- <. ( Base ` ndx ) , T >. = <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >.
16		eqid 2622	. . . . . . 7 |- ( f o. g ) = ( f o. g )
17	14, 14, 16	mpt2eq123i 6718	. . . . . 6 |- ( f e. T , g e. T |-> ( f o. g ) ) = ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) )
18	17	opeq2i 4406	. . . . 5 |- <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. = <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >.
19	15, 18	preq12i 4273	. . . 4 |- { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } = { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. }
20	13, 19	syl6eqr 2674	. . 3 |- ( W e. H -> ( ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) ` W ) = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } )
21	4, 20	sylan9eq 2676	. 2 |- ( ( K e. V /\ W e. H ) -> ( ( TGrp ` K ) ` W ) = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } )
22	1, 21	syl5eq 2668	1 |- ( ( K e. V /\ W e. H ) -> G = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cpr 4179   <.cop 4183   |-> cmpt 4729   o. ccom 5118   ` cfv 5888   |-> cmpt2 6652   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   LHypclh 35270   LTrncltrn 35387   TGrpctgrp 36030

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-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-pr 4906

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-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-oprab 6654  df-mpt2 6655  df-tgrp 36031

This theorem is referenced by:  tgrpbase  36034  tgrpopr  36035  dvaabl  36313