Theorem lcfrlem16 36847

Description: Lemma for lcfr 36874. (Contributed by NM, 8-Mar-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 ) )
lcfrlem16.p	|- P = ( LSubSp ` D )
lcfrlem16.g	|- ( ph -> G e. P )
lcfrlem16.gs	|- ( ph -> G C_ C )
lcfrlem16.m	|- E = U_ g e. G ( ._|_ ` ( L ` g ) )
lcfrlem16.x	|- ( ph -> X e. ( E \ { .0. } ) )

Assertion

Ref	Expression
lcfrlem16	|- ( ph -> ( J ` X ) e. G )

Distinct variable groups:   x, w, ._|_   x, .0.   x, v, V   x, .x.   v, k, w, x, X   x, .+   x, R   f, k, v, w, .+   f, F, k   g, k, G   f, g, J, k   f, L, k   ._|_ , f, k, v   R, f, k, v   S, k   .x. , f, k, v, w   U, k   f, V, g, x   f, X   v, g, w, x, X   ph, g, k

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

Proof of Theorem lcfrlem16

Step	Hyp	Ref	Expression
1		lcfrlem16.x	. . . . 5 |- ( ph -> X e. ( E \ { .0. } ) )
2	1	eldifad 3586	. . . 4 |- ( ph -> X e. E )
3		lcfrlem16.m	. . . 4 |- E = U_ g e. G ( ._|_ ` ( L ` g ) )
4	2, 3	syl6eleq 2711	. . 3 |- ( ph -> X e. U_ g e. G ( ._|_ ` ( L ` g ) ) )
5		eliun 4524	. . 3 |- ( X e. U_ g e. G ( ._|_ ` ( L ` g ) ) <-> E. g e. G X e. ( ._|_ ` ( L ` g ) ) )
6	4, 5	sylib 208	. 2 |- ( ph -> E. g e. G X e. ( ._|_ ` ( L ` g ) ) )
7		lcf1o.s	. . . . 5 |- S = (Scalar ` U )
8		lcf1o.r	. . . . 5 |- R = ( Base ` S )
9		lcf1o.f	. . . . 5 |- F = (LFnl ` U )
10		lcf1o.l	. . . . 5 |- L = (LKer ` U )
11		lcf1o.d	. . . . 5 |- D = (LDual ` U )
12		eqid 2622	. . . . 5 |- ( .s ` D ) = ( .s ` D )
13		lcf1o.h	. . . . . . 7 |- H = ( LHyp ` K )
14		lcf1o.u	. . . . . . 7 |- U = ( ( DVecH ` K ) ` W )
15		lcflo.k	. . . . . . 7 |- ( ph -> ( K e. HL /\ W e. H ) )
16	13, 14, 15	dvhlvec 36398	. . . . . 6 |- ( ph -> U e. LVec )
17	16	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> U e. LVec )
18		lcfrlem16.g	. . . . . . . 8 |- ( ph -> G e. P )
19		eqid 2622	. . . . . . . . 9 |- ( Base ` D ) = ( Base ` D )
20		lcfrlem16.p	. . . . . . . . 9 |- P = ( LSubSp ` D )
21	19, 20	lssel 18938	. . . . . . . 8 |- ( ( G e. P /\ g e. G ) -> g e. ( Base ` D ) )
22	18, 21	sylan 488	. . . . . . 7 |- ( ( ph /\ g e. G ) -> g e. ( Base ` D ) )
23	13, 14, 15	dvhlmod 36399	. . . . . . . . 9 |- ( ph -> U e. LMod )
24	9, 11, 19, 23	ldualvbase 34413	. . . . . . . 8 |- ( ph -> ( Base ` D ) = F )
25	24	adantr 481	. . . . . . 7 |- ( ( ph /\ g e. G ) -> ( Base ` D ) = F )
26	22, 25	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ g e. G ) -> g e. F )
27	26	3adant3 1081	. . . . 5 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> g e. F )
28		lcf1o.o	. . . . . . 7 |- ._|_ = ( ( ocH ` K ) ` W )
29		lcf1o.v	. . . . . . 7 |- V = ( Base ` U )
30		lcf1o.a	. . . . . . 7 |- .+ = ( +g ` U )
31		lcf1o.t	. . . . . . 7 |- .x. = ( .s ` U )
32		lcf1o.z	. . . . . . 7 |- .0. = ( 0g ` U )
33		lcf1o.q	. . . . . . 7 |- Q = ( 0g ` D )
34		lcf1o.c	. . . . . . 7 |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
35		lcf1o.j	. . . . . . 7 |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
36	15	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ g e. G ) -> ( K e. HL /\ W e. H ) )
37	23	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ g e. G ) -> U e. LMod )
38	29, 9, 10, 37, 26	lkrssv 34383	. . . . . . . . . . . . 13 |- ( ( ph /\ g e. G ) -> ( L ` g ) C_ V )
39	13, 14, 29, 28	dochssv 36644	. . . . . . . . . . . . 13 |- ( ( ( K e. HL /\ W e. H ) /\ ( L ` g ) C_ V ) -> ( ._|_ ` ( L ` g ) ) C_ V )
40	36, 38, 39	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ g e. G ) -> ( ._|_ ` ( L ` g ) ) C_ V )
41	40	ralrimiva 2966	. . . . . . . . . . 11 |- ( ph -> A. g e. G ( ._|_ ` ( L ` g ) ) C_ V )
42		iunss 4561	. . . . . . . . . . 11 |- ( U_ g e. G ( ._|_ ` ( L ` g ) ) C_ V <-> A. g e. G ( ._|_ ` ( L ` g ) ) C_ V )
43	41, 42	sylibr 224	. . . . . . . . . 10 |- ( ph -> U_ g e. G ( ._|_ ` ( L ` g ) ) C_ V )
44	3, 43	syl5eqss 3649	. . . . . . . . 9 |- ( ph -> E C_ V )
45	44	ssdifd 3746	. . . . . . . 8 |- ( ph -> ( E \ { .0. } ) C_ ( V \ { .0. } ) )
46	45, 1	sseldd 3604	. . . . . . 7 |- ( ph -> X e. ( V \ { .0. } ) )
47	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem10 36841	. . . . . 6 |- ( ph -> ( J ` X ) e. F )
48	47	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( J ` X ) e. F )
49		eqid 2622	. . . . . . 7 |- (LSAtoms ` U ) = (LSAtoms ` U )
50	15	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( K e. HL /\ W e. H ) )
51		simp3 1063	. . . . . . . . 9 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> X e. ( ._|_ ` ( L ` g ) ) )
52		eldifsni 4320	. . . . . . . . . . 11 |- ( X e. ( E \ { .0. } ) -> X =/= .0. )
53	1, 52	syl 17	. . . . . . . . . 10 |- ( ph -> X =/= .0. )
54	53	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> X =/= .0. )
55		eldifsn 4317	. . . . . . . . 9 |- ( X e. ( ( ._|_ ` ( L ` g ) ) \ { .0. } ) <-> ( X e. ( ._|_ ` ( L ` g ) ) /\ X =/= .0. ) )
56	51, 54, 55	sylanbrc 698	. . . . . . . 8 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> X e. ( ( ._|_ ` ( L ` g ) ) \ { .0. } ) )
57	13, 28, 14, 29, 32, 9, 10, 50, 27, 56, 49	dochsnkrlem2 36759	. . . . . . 7 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( ._|_ ` ( L ` g ) ) e. (LSAtoms ` U ) )
58	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem15 36846	. . . . . . . . . 10 |- ( ph -> X e. ( ._|_ ` ( L ` ( J ` X ) ) ) )
59		eldifsn 4317	. . . . . . . . . 10 |- ( X e. ( ( ._|_ ` ( L ` ( J ` X ) ) ) \ { .0. } ) <-> ( X e. ( ._|_ ` ( L ` ( J ` X ) ) ) /\ X =/= .0. ) )
60	58, 53, 59	sylanbrc 698	. . . . . . . . 9 |- ( ph -> X e. ( ( ._|_ ` ( L ` ( J ` X ) ) ) \ { .0. } ) )
61	13, 28, 14, 29, 32, 9, 10, 15, 47, 60, 49	dochsnkrlem2 36759	. . . . . . . 8 |- ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) e. (LSAtoms ` U ) )
62	61	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( ._|_ ` ( L ` ( J ` X ) ) ) e. (LSAtoms ` U ) )
63	58	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> X e. ( ._|_ ` ( L ` ( J ` X ) ) ) )
64	32, 49, 17, 57, 62, 54, 51, 63	lsat2el 34294	. . . . . 6 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( ._|_ ` ( L ` g ) ) = ( ._|_ ` ( L ` ( J ` X ) ) ) )
65		eqid 2622	. . . . . . 7 |- ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
66		lcfrlem16.gs	. . . . . . . . . 10 |- ( ph -> G C_ C )
67	66	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> G C_ C )
68		simp2 1062	. . . . . . . . 9 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> g e. G )
69	67, 68	sseldd 3604	. . . . . . . 8 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> g e. C )
70	13, 65, 28, 14, 9, 10, 34, 50, 27	lcfl5 36785	. . . . . . . 8 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( g e. C <-> ( L ` g ) e. ran ( ( DIsoH ` K ) ` W ) ) )
71	69, 70	mpbid 222	. . . . . . 7 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( L ` g ) e. ran ( ( DIsoH ` K ) ` W ) )
72	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem13 36844	. . . . . . . . . 10 |- ( ph -> ( J ` X ) e. ( C \ { Q } ) )
73	72	eldifad 3586	. . . . . . . . 9 |- ( ph -> ( J ` X ) e. C )
74	13, 65, 28, 14, 9, 10, 34, 15, 47	lcfl5 36785	. . . . . . . . 9 |- ( ph -> ( ( J ` X ) e. C <-> ( L ` ( J ` X ) ) e. ran ( ( DIsoH ` K ) ` W ) ) )
75	73, 74	mpbid 222	. . . . . . . 8 |- ( ph -> ( L ` ( J ` X ) ) e. ran ( ( DIsoH ` K ) ` W ) )
76	75	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( L ` ( J ` X ) ) e. ran ( ( DIsoH ` K ) ` W ) )
77	13, 65, 28, 50, 71, 76	doch11 36662	. . . . . 6 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( ( ._|_ ` ( L ` g ) ) = ( ._|_ ` ( L ` ( J ` X ) ) ) <-> ( L ` g ) = ( L ` ( J ` X ) ) ) )
78	64, 77	mpbid 222	. . . . 5 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( L ` g ) = ( L ` ( J ` X ) ) )
79	7, 8, 9, 10, 11, 12, 17, 27, 48, 78	eqlkr4 34452	. . . 4 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> E. k e. R ( J ` X ) = ( k ( .s ` D ) g ) )
80	23	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> U e. LMod )
81	80	adantr 481	. . . . . . 7 |- ( ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) /\ k e. R ) -> U e. LMod )
82	18	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> G e. P )
83	82	adantr 481	. . . . . . 7 |- ( ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) /\ k e. R ) -> G e. P )
84		simpr 477	. . . . . . 7 |- ( ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) /\ k e. R ) -> k e. R )
85		simpl2 1065	. . . . . . 7 |- ( ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) /\ k e. R ) -> g e. G )
86	7, 8, 11, 12, 20, 81, 83, 84, 85	ldualssvscl 34445	. . . . . 6 |- ( ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) /\ k e. R ) -> ( k ( .s ` D ) g ) e. G )
87		eleq1 2689	. . . . . 6 |- ( ( J ` X ) = ( k ( .s ` D ) g ) -> ( ( J ` X ) e. G <-> ( k ( .s ` D ) g ) e. G ) )
88	86, 87	syl5ibrcom 237	. . . . 5 |- ( ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) /\ k e. R ) -> ( ( J ` X ) = ( k ( .s ` D ) g ) -> ( J ` X ) e. G ) )
89	88	rexlimdva 3031	. . . 4 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( E. k e. R ( J ` X ) = ( k ( .s ` D ) g ) -> ( J ` X ) e. G ) )
90	79, 89	mpd 15	. . 3 |- ( ( ph /\ g e. G /\ X e. ( ._|_ ` ( L ` g ) ) ) -> ( J ` X ) e. G )
91	90	rexlimdv3a 3033	. 2 |- ( ph -> ( E. g e. G X e. ( ._|_ ` ( L ` g ) ) -> ( J ` X ) e. G ) )
92	6, 91	mpd 15	1 |- ( ph -> ( J ` X ) e. G )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   {csn 4177   U_ciun 4520   |-> cmpt 4729   ran crn 5115   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LModclmod 18863   LSubSpclss 18932   LVecclvec 19102  LSAtomsclsa 34261  LFnlclfn 34344  LKerclk 34372  LDualcld 34410   HLchlt 34637   LHypclh 35270   DVecHcdvh 36367   DIsoHcdih 36517   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-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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-iin 4523  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  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-dvr 18683  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lsatoms 34263  df-lshyp 34264  df-lfl 34345  df-lkr 34373  df-ldual 34411  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tgrp 36031  df-tendo 36043  df-edring 36045  df-dveca 36291  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518  df-doch 36637  df-djh 36684

This theorem is referenced by:  lcfrlem27  36858  lcfrlem37  36868