Theorem lcfrlem8 37419

Description: Lemma for lcf1o 37421 and lcfr 37455. (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 4042	. . . . . . 7 |- ( x = X -> { x } = { X } )
3	2	fveq2d 5876	. . . . . 6 |- ( x = X -> ( ._|_ ` { x } ) = ( ._|_ ` { X } ) )
4		oveq2 6304	. . . . . . . 8 |- ( x = X -> ( k .x. x ) = ( k .x. X ) )
5	4	oveq2d 6312	. . . . . . 7 |- ( x = X -> ( w .+ ( k .x. x ) ) = ( w .+ ( k .x. X ) ) )
6	5	eqeq2d 2471	. . . . . 6 |- ( x = X -> ( v = ( w .+ ( k .x. x ) ) <-> v = ( w .+ ( k .x. X ) ) ) )
7	3, 6	rexeqbidv 3069	. . . . 5 |- ( x = X -> ( E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) <-> E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
8	7	riotabidv 6260	. . . 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 4544	. . 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 5882	. . . . 5 |- ( Base ` U ) e. _V
13	11, 12	eqeltri 2541	. . . 4 |- V e. _V
14	13	mptex 6144	. . 3 |- ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) e. _V
15	9, 10, 14	fvmpt 5956	. 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 1395   e. wcel 1819   E.wrex 2808   {crab 2811   _Vcvv 3109   \ cdif 3468   {csn 4032   |-> cmpt 4515   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   +g cplusg 14712  Scalarcsca 14715   .scvsca 14716   0gc0g 14857  LFnlclfn 34925  LKerclk 34953  LDualcld 34991   HLchlt 35218   LHypclh 35851   DVecHcdvh 36948   ocHcoch 37217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299

This theorem is referenced by:  lcfrlem9  37420  lcfrlem10  37422  lcfrlem11  37423