Theorem lshpkrlem3 34399

Description: Lemma for lshpkrex 34405. Defining property of G ` X. (Contributed by NM, 15-Jul-2014.)

Hypotheses

Ref	Expression
lshpkrlem.v	|- V = ( Base ` W )
lshpkrlem.a	|- .+ = ( +g ` W )
lshpkrlem.n	|- N = ( LSpan ` W )
lshpkrlem.p	|- .(+) = ( LSSum ` W )
lshpkrlem.h	|- H = (LSHyp ` W )
lshpkrlem.w	|- ( ph -> W e. LVec )
lshpkrlem.u	|- ( ph -> U e. H )
lshpkrlem.z	|- ( ph -> Z e. V )
lshpkrlem.x	|- ( ph -> X e. V )
lshpkrlem.e	|- ( ph -> ( U .(+) ( N ` { Z } ) ) = V )
lshpkrlem.d	|- D = (Scalar ` W )
lshpkrlem.k	|- K = ( Base ` D )
lshpkrlem.t	|- .x. = ( .s ` W )
lshpkrlem.o	|- .0. = ( 0g ` D )
lshpkrlem.g	|- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) )

Assertion

Ref	Expression
lshpkrlem3	|- ( ph -> E. z e. U X = ( z .+ ( ( G ` X ) .x. Z ) ) )

Distinct variable groups:   x, k, y, .+   k, K, x   .0. , k   .x. , k, x, y   U, k, x, y   x, V   k, X, x, y   k, Z, x, y   z, .+   z, G   z, U   z, X   z, Z, k, x, y   z, .x.

Allowed substitution hints:   ph( x, y, z, k)   D( x, y, z, k)   .(+) ( x, y, z, k)   G( x, y, k)   H( x, y, z, k)   K( y, z)   N( x, y, z, k)   V( y, z, k)   W( x, y, z, k)   .0. ( x, y, z)

Proof of Theorem lshpkrlem3

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpkrlem.v	. . . . 5 |- V = ( Base ` W )
2		lshpkrlem.a	. . . . 5 |- .+ = ( +g ` W )
3		lshpkrlem.n	. . . . 5 |- N = ( LSpan ` W )
4		lshpkrlem.p	. . . . 5 |- .(+) = ( LSSum ` W )
5		lshpkrlem.h	. . . . 5 |- H = (LSHyp ` W )
6		lshpkrlem.w	. . . . 5 |- ( ph -> W e. LVec )
7		lshpkrlem.u	. . . . 5 |- ( ph -> U e. H )
8		lshpkrlem.z	. . . . 5 |- ( ph -> Z e. V )
9		lshpkrlem.x	. . . . 5 |- ( ph -> X e. V )
10		lshpkrlem.e	. . . . 5 |- ( ph -> ( U .(+) ( N ` { Z } ) ) = V )
11		lshpkrlem.d	. . . . 5 |- D = (Scalar ` W )
12		lshpkrlem.k	. . . . 5 |- K = ( Base ` D )
13		lshpkrlem.t	. . . . 5 |- .x. = ( .s ` W )
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	lshpsmreu 34396	. . . 4 |- ( ph -> E! l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) )
15		riotasbc 6626	. . . 4 |- ( E! l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) -> [. ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) )
16	14, 15	syl 17	. . 3 |- ( ph -> [. ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) )
17		eqeq1 2626	. . . . . . 7 |- ( x = X -> ( x = ( z .+ ( l .x. Z ) ) <-> X = ( z .+ ( l .x. Z ) ) ) )
18	17	rexbidv 3052	. . . . . 6 |- ( x = X -> ( E. z e. U x = ( z .+ ( l .x. Z ) ) <-> E. z e. U X = ( z .+ ( l .x. Z ) ) ) )
19	18	riotabidv 6613	. . . . 5 |- ( x = X -> ( iota_ l e. K E. z e. U x = ( z .+ ( l .x. Z ) ) ) = ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) )
20		lshpkrlem.g	. . . . . 6 |- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) )
21		oveq1 6657	. . . . . . . . . . . 12 |- ( k = l -> ( k .x. Z ) = ( l .x. Z ) )
22	21	oveq2d 6666	. . . . . . . . . . 11 |- ( k = l -> ( y .+ ( k .x. Z ) ) = ( y .+ ( l .x. Z ) ) )
23	22	eqeq2d 2632	. . . . . . . . . 10 |- ( k = l -> ( x = ( y .+ ( k .x. Z ) ) <-> x = ( y .+ ( l .x. Z ) ) ) )
24	23	rexbidv 3052	. . . . . . . . 9 |- ( k = l -> ( E. y e. U x = ( y .+ ( k .x. Z ) ) <-> E. y e. U x = ( y .+ ( l .x. Z ) ) ) )
25		oveq1 6657	. . . . . . . . . . 11 |- ( y = z -> ( y .+ ( l .x. Z ) ) = ( z .+ ( l .x. Z ) ) )
26	25	eqeq2d 2632	. . . . . . . . . 10 |- ( y = z -> ( x = ( y .+ ( l .x. Z ) ) <-> x = ( z .+ ( l .x. Z ) ) ) )
27	26	cbvrexv 3172	. . . . . . . . 9 |- ( E. y e. U x = ( y .+ ( l .x. Z ) ) <-> E. z e. U x = ( z .+ ( l .x. Z ) ) )
28	24, 27	syl6bb 276	. . . . . . . 8 |- ( k = l -> ( E. y e. U x = ( y .+ ( k .x. Z ) ) <-> E. z e. U x = ( z .+ ( l .x. Z ) ) ) )
29	28	cbvriotav 6622	. . . . . . 7 |- ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) = ( iota_ l e. K E. z e. U x = ( z .+ ( l .x. Z ) ) )
30	29	mpteq2i 4741	. . . . . 6 |- ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) ) = ( x e. V |-> ( iota_ l e. K E. z e. U x = ( z .+ ( l .x. Z ) ) ) )
31	20, 30	eqtri 2644	. . . . 5 |- G = ( x e. V |-> ( iota_ l e. K E. z e. U x = ( z .+ ( l .x. Z ) ) ) )
32		riotaex 6615	. . . . 5 |- ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) e. _V
33	19, 31, 32	fvmpt 6282	. . . 4 |- ( X e. V -> ( G ` X ) = ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) )
34		dfsbcq 3437	. . . 4 |- ( ( G ` X ) = ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) -> ( [. ( G ` X ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) <-> [. ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) ) )
35	9, 33, 34	3syl 18	. . 3 |- ( ph -> ( [. ( G ` X ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) <-> [. ( iota_ l e. K E. z e. U X = ( z .+ ( l .x. Z ) ) ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) ) )
36	16, 35	mpbird 247	. 2 |- ( ph -> [. ( G ` X ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) )
37		fvex 6201	. . 3 |- ( G ` X ) e. _V
38		oveq1 6657	. . . . . 6 |- ( l = ( G ` X ) -> ( l .x. Z ) = ( ( G ` X ) .x. Z ) )
39	38	oveq2d 6666	. . . . 5 |- ( l = ( G ` X ) -> ( z .+ ( l .x. Z ) ) = ( z .+ ( ( G ` X ) .x. Z ) ) )
40	39	eqeq2d 2632	. . . 4 |- ( l = ( G ` X ) -> ( X = ( z .+ ( l .x. Z ) ) <-> X = ( z .+ ( ( G ` X ) .x. Z ) ) ) )
41	40	rexbidv 3052	. . 3 |- ( l = ( G ` X ) -> ( E. z e. U X = ( z .+ ( l .x. Z ) ) <-> E. z e. U X = ( z .+ ( ( G ` X ) .x. Z ) ) ) )
42	37, 41	sbcie 3470	. 2 |- ( [. ( G ` X ) / l ]. E. z e. U X = ( z .+ ( l .x. Z ) ) <-> E. z e. U X = ( z .+ ( ( G ` X ) .x. Z ) ) )
43	36, 42	sylib 208	1 |- ( ph -> E. z e. U X = ( z .+ ( ( G ` X ) .x. Z ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   E!wreu 2914   [.wsbc 3435   {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   LSSumclsm 18049   LSpanclspn 18971   LVecclvec 19102  LSHypclsh 34262

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lshyp 34264

This theorem is referenced by:  lshpkrlem6  34402