Theorem lcfrlem8 35162

Description: Lemma for lcf1o 35164 and lcfr 35198. (Contributed by NM, 21-Feb-2015.)

Hypotheses

Ref	Expression
lcf1o.h	|- H = ( LHyp ` K )
lcf1o.o	|- ._|_ = ( ( ocH ` K ) ` W )
lcf1o.u	|- U = ( ( DVecH ` K ) ` W )
lcf1o.v	|- V = ( Base ` U )
lcf1o.a	|- .+ = ( +g ` U )
lcf1o.t	|- .x. = ( .s ` U )
lcf1o.s	|- S = (Scalar ` U )
lcf1o.r	|- R = ( Base ` S )
lcf1o.z	|- .0. = ( 0g ` U )
lcf1o.f	|- F = (LFnl ` U )
lcf1o.l	|- L = (LKer ` U )
lcf1o.d	|- D = (LDual ` U )
lcf1o.q	|- Q = ( 0g ` D )
lcf1o.c	|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
lcf1o.j	|- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
lcflo.k	|- ( ph -> ( K e. HL /\ W e. H ) )
lcfrlem8.x	|- ( ph -> X e. ( V \ { .0. } ) )

Assertion

Ref	Expression
lcfrlem8	|- ( ph -> ( J ` X ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )

Distinct variable groups:   x, w, ._|_   x, .0.   x, v, V   x, .x.   v, k, w, x, X   x, .+   x, R

Allowed substitution hints:   ph( x, w, v, f, k)   C( x, w, v, f, k)   D( x, w, v, f, k)   .+ ( w, v, f, k)   Q( x, w, v, f, k)   R( w, v, f, k)   S( x, w, v, f, k)   .x. ( w, v, f, k)   U( x, w, v, f, k)   F( x, w, v, f, k)   H( x, w, v, f, k)   J( x, w, v, f, k)   K( x, w, v, f, k)   L( x, w, v, f, k)   ._|_ ( v, f, k)   V( w, f, k)   W( x, w, v, f, k)   X( f)   .0. ( w, v, f, k)

Proof of Theorem lcfrlem8

Step	Hyp	Ref	Expression
1		lcfrlem8.x	. 2 |- ( ph -> X e. ( V \ { .0. } ) )
2		sneq 3990	. . . . . . 7 |- ( x = X -> { x } = { X } )
3	2	fveq2d 5892	. . . . . 6 |- ( x = X -> ( ._|_ ` { x } ) = ( ._|_ ` { X } ) )
4		oveq2 6323	. . . . . . . 8 |- ( x = X -> ( k .x. x ) = ( k .x. X ) )
5	4	oveq2d 6331	. . . . . . 7 |- ( x = X -> ( w .+ ( k .x. x ) ) = ( w .+ ( k .x. X ) ) )
6	5	eqeq2d 2472	. . . . . 6 |- ( x = X -> ( v = ( w .+ ( k .x. x ) ) <-> v = ( w .+ ( k .x. X ) ) ) )
7	3, 6	rexeqbidv 3014	. . . . 5 |- ( x = X -> ( E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) <-> E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
8	7	riotabidv 6279	. . . 4 |- ( x = X -> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) = ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
9	8	mpteq2dv 4504	. . 3 |- ( x = X -> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )
10		lcf1o.j	. . 3 |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
11		lcf1o.v	. . . . 5 |- V = ( Base ` U )
12		fvex 5898	. . . . 5 |- ( Base ` U ) e. _V
13	11, 12	eqeltri 2536	. . . 4 |- V e. _V
14	13	mptex 6161	. . 3 |- ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) e. _V
15	9, 10, 14	fvmpt 5971	. 2 |- ( X e. ( V \ { .0. } ) -> ( J ` X ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )
16	1, 15	syl 17	1 |- ( ph -> ( J ` X ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1455   e. wcel 1898   E.wrex 2750   {crab 2753   _Vcvv 3057   \ cdif 3413   {csn 3980   |-> cmpt 4475   ` cfv 5601   iota_crio 6276  (class class class)co 6315   Basecbs 15170   +g cplusg 15239  Scalarcsca 15242   .scvsca 15243   0gc0g 15387  LFnlclfn 32668  LKerclk 32696  LDualcld 32734   HLchlt 32961   LHypclh 33594   DVecHcdvh 34691   ocHcoch 34960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318

This theorem is referenced by:  lcfrlem9  35163  lcfrlem10  35165  lcfrlem11  35166