Theorem istrnN 35444

Description: The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
trnset.a	|- A = ( Atoms ` K )
trnset.s	|- S = ( PSubSp ` K )
trnset.p	|- .+ = ( +P ` K )
trnset.o	|- ._|_ = ( _|_P ` K )
trnset.w	|- W = ( WAtoms ` K )
trnset.m	|- M = ( PAut ` K )
trnset.l	|- L = ( Dil ` K )
trnset.t	|- T = ( Trn ` K )

Assertion

Ref	Expression
istrnN	|- ( ( K e. B /\ D e. A ) -> ( F e. ( T ` D ) <-> ( F e. ( L ` D ) /\ A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( F ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( F ` r ) ) i^i ( ._|_ ` { D } ) ) ) ) )

Distinct variable groups:   r, q, K   W, q, r   D, q, r   F, q, r

Allowed substitution hints:   A( r, q)   B( r, q)   .+ ( r, q)   S( r, q)   T( r, q)   L( r, q)   M( r, q)   ._|_ ( r, q)

Proof of Theorem istrnN

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trnset.a	. . . 4 |- A = ( Atoms ` K )
2		trnset.s	. . . 4 |- S = ( PSubSp ` K )
3		trnset.p	. . . 4 |- .+ = ( +P ` K )
4		trnset.o	. . . 4 |- ._|_ = ( _|_P ` K )
5		trnset.w	. . . 4 |- W = ( WAtoms ` K )
6		trnset.m	. . . 4 |- M = ( PAut ` K )
7		trnset.l	. . . 4 |- L = ( Dil ` K )
8		trnset.t	. . . 4 |- T = ( Trn ` K )
9	1, 2, 3, 4, 5, 6, 7, 8	trnsetN 35443	. . 3 |- ( ( K e. B /\ D e. A ) -> ( T ` D ) = { f e. ( L ` D ) | A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { D } ) ) } )
10	9	eleq2d 2687	. 2 |- ( ( K e. B /\ D e. A ) -> ( F e. ( T ` D ) <-> F e. { f e. ( L ` D ) | A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { D } ) ) } ) )
11		fveq1 6190	. . . . . . 7 |- ( f = F -> ( f ` q ) = ( F ` q ) )
12	11	oveq2d 6666	. . . . . 6 |- ( f = F -> ( q .+ ( f ` q ) ) = ( q .+ ( F ` q ) ) )
13	12	ineq1d 3813	. . . . 5 |- ( f = F -> ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( q .+ ( F ` q ) ) i^i ( ._|_ ` { D } ) ) )
14		fveq1 6190	. . . . . . 7 |- ( f = F -> ( f ` r ) = ( F ` r ) )
15	14	oveq2d 6666	. . . . . 6 |- ( f = F -> ( r .+ ( f ` r ) ) = ( r .+ ( F ` r ) ) )
16	15	ineq1d 3813	. . . . 5 |- ( f = F -> ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( F ` r ) ) i^i ( ._|_ ` { D } ) ) )
17	13, 16	eqeq12d 2637	. . . 4 |- ( f = F -> ( ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { D } ) ) <-> ( ( q .+ ( F ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( F ` r ) ) i^i ( ._|_ ` { D } ) ) ) )
18	17	2ralbidv 2989	. . 3 |- ( f = F -> ( A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { D } ) ) <-> A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( F ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( F ` r ) ) i^i ( ._|_ ` { D } ) ) ) )
19	18	elrab 3363	. 2 |- ( F e. { f e. ( L ` D ) | A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { D } ) ) } <-> ( F e. ( L ` D ) /\ A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( F ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( F ` r ) ) i^i ( ._|_ ` { D } ) ) ) )
20	10, 19	syl6bb 276	1 |- ( ( K e. B /\ D e. A ) -> ( F e. ( T ` D ) <-> ( F e. ( L ` D ) /\ A. q e. ( W ` D ) A. r e. ( W ` D ) ( ( q .+ ( F ` q ) ) i^i ( ._|_ ` { D } ) ) = ( ( r .+ ( F ` r ) ) i^i ( ._|_ ` { D } ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   i^i cin 3573   {csn 4177   ` cfv 5888  (class class class)co 6650   Atomscatm 34550   PSubSpcpsubsp 34782   +Pcpadd 35081   _|_PcpolN 35188   WAtomscwpointsN 35272   PAutcpautN 35273   DilcdilN 35388   TrnctrnN 35389

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-ov 6653  df-trnN 35393

This theorem is referenced by: (None)