Theorem lcfrlem8 35290

Description: Lemma for lcf1o 35292 and lcfr 35326. (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 3908	. . . . . . 7 |- ( x = X -> { x } = { X } )
3	2	fveq2d 5716	. . . . . 6 |- ( x = X -> ( ._|_ ` { x } ) = ( ._|_ ` { X } ) )
4		oveq2 6120	. . . . . . . 8 |- ( x = X -> ( k .x. x ) = ( k .x. X ) )
5	4	oveq2d 6128	. . . . . . 7 |- ( x = X -> ( w .+ ( k .x. x ) ) = ( w .+ ( k .x. X ) ) )
6	5	eqeq2d 2454	. . . . . 6 |- ( x = X -> ( v = ( w .+ ( k .x. x ) ) <-> v = ( w .+ ( k .x. X ) ) ) )
7	3, 6	rexeqbidv 2953	. . . . 5 |- ( x = X -> ( E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) <-> E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
8	7	riotabidv 6075	. . . 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 4400	. . 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 5722	. . . . 5 |- ( Base ` U ) e. _V
13	11, 12	eqeltri 2513	. . . 4 |- V e. _V
14	13	mptex 5969	. . 3 |- ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) e. _V
15	9, 10, 14	fvmpt 5795	. 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 16	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 369   = wceq 1369   e. wcel 1756   E.wrex 2737   {crab 2740   _Vcvv 2993   \ cdif 3346   {csn 3898   e. cmpt 4371   ` cfv 5439   iota_crio 6072  (class class class)co 6112   Basecbs 14195   +g cplusg 14259  Scalarcsca 14262   .scvsca 14263   0gc0g 14399  LFnlclfn 32798  LKerclk 32826  LDualcld 32864   HLchlt 33091   LHypclh 33724   DVecHcdvh 34819   ocHcoch 35088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pr 4552

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115

This theorem is referenced by:  lcfrlem9  35291  lcfrlem10  35293  lcfrlem11  35294