Theorem lcfrlem8 36838

Description: Lemma for lcf1o 36840 and lcfr 36874. (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 4187	. . . . . . 7 |- ( x = X -> { x } = { X } )
3	2	fveq2d 6195	. . . . . 6 |- ( x = X -> ( ._|_ ` { x } ) = ( ._|_ ` { X } ) )
4		oveq2 6658	. . . . . . . 8 |- ( x = X -> ( k .x. x ) = ( k .x. X ) )
5	4	oveq2d 6666	. . . . . . 7 |- ( x = X -> ( w .+ ( k .x. x ) ) = ( w .+ ( k .x. X ) ) )
6	5	eqeq2d 2632	. . . . . 6 |- ( x = X -> ( v = ( w .+ ( k .x. x ) ) <-> v = ( w .+ ( k .x. X ) ) ) )
7	3, 6	rexeqbidv 3153	. . . . 5 |- ( x = X -> ( E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) <-> E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
8	7	riotabidv 6613	. . . 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 4745	. . 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 6201	. . . . 5 |- ( Base ` U ) e. _V
13	11, 12	eqeltri 2697	. . . 4 |- V e. _V
14	13	mptex 6486	. . 3 |- ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) e. _V
15	9, 10, 14	fvmpt 6282	. 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 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100  LFnlclfn 34344  LKerclk 34372  LDualcld 34410   HLchlt 34637   LHypclh 35270   DVecHcdvh 36367   ocHcoch 36636

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-riota 6611  df-ov 6653

This theorem is referenced by:  lcfrlem9  36839  lcfrlem10  36841  lcfrlem11  36842