Theorem diaelrnN 36334

Description: Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
diaelrn.h	|- H = ( LHyp ` K )
diaelrn.t	|- T = ( ( LTrn ` K ) ` W )
diaelrn.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

Ref	Expression
diaelrnN	|- ( ( ( K e. V /\ W e. H ) /\ S e. ran I ) -> S C_ T )

Proof of Theorem diaelrnN

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` K ) = ( Base ` K )
2		eqid 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
3		diaelrn.h	. . . . 5 |- H = ( LHyp ` K )
4		diaelrn.i	. . . . 5 |- I = ( ( DIsoA ` K ) ` W )
5	1, 2, 3, 4	diafn 36323	. . . 4 |- ( ( K e. V /\ W e. H ) -> I Fn { y e. ( Base ` K ) | y ( le ` K ) W } )
6		fvelrnb 6243	. . . 4 |- ( I Fn { y e. ( Base ` K ) | y ( le ` K ) W } -> ( S e. ran I <-> E. x e. { y e. ( Base ` K ) | y ( le ` K ) W } ( I ` x ) = S ) )
7	5, 6	syl 17	. . 3 |- ( ( K e. V /\ W e. H ) -> ( S e. ran I <-> E. x e. { y e. ( Base ` K ) | y ( le ` K ) W } ( I ` x ) = S ) )
8		breq1 4656	. . . . . 6 |- ( y = x -> ( y ( le ` K ) W <-> x ( le ` K ) W ) )
9	8	elrab 3363	. . . . 5 |- ( x e. { y e. ( Base ` K ) | y ( le ` K ) W } <-> ( x e. ( Base ` K ) /\ x ( le ` K ) W ) )
10		diaelrn.t	. . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
11	1, 2, 3, 10, 4	diass 36331	. . . . . . 7 |- ( ( ( K e. V /\ W e. H ) /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) ) -> ( I ` x ) C_ T )
12	11	ex 450	. . . . . 6 |- ( ( K e. V /\ W e. H ) -> ( ( x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( I ` x ) C_ T ) )
13		sseq1 3626	. . . . . . 7 |- ( ( I ` x ) = S -> ( ( I ` x ) C_ T <-> S C_ T ) )
14	13	biimpcd 239	. . . . . 6 |- ( ( I ` x ) C_ T -> ( ( I ` x ) = S -> S C_ T ) )
15	12, 14	syl6 35	. . . . 5 |- ( ( K e. V /\ W e. H ) -> ( ( x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( ( I ` x ) = S -> S C_ T ) ) )
16	9, 15	syl5bi 232	. . . 4 |- ( ( K e. V /\ W e. H ) -> ( x e. { y e. ( Base ` K ) | y ( le ` K ) W } -> ( ( I ` x ) = S -> S C_ T ) ) )
17	16	rexlimdv 3030	. . 3 |- ( ( K e. V /\ W e. H ) -> ( E. x e. { y e. ( Base ` K ) | y ( le ` K ) W } ( I ` x ) = S -> S C_ T ) )
18	7, 17	sylbid 230	. 2 |- ( ( K e. V /\ W e. H ) -> ( S e. ran I -> S C_ T ) )
19	18	imp 445	1 |- ( ( ( K e. V /\ W e. H ) /\ S e. ran I ) -> S C_ T )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   class class class wbr 4653   ran crn 5115   Fn wfn 5883   ` cfv 5888   Basecbs 15857   lecple 15948   LHypclh 35270   LTrncltrn 35387   DIsoAcdia 36317

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-disoa 36318

This theorem is referenced by:  dvadiaN  36417  djaclN  36425  djajN  36426