Theorem lcfrlem2 35123

Description: Lemma for lcfr 35165. (Contributed by NM, 27-Feb-2015.)

Hypotheses

Ref	Expression
lcfrlem1.v	|- V = ( Base ` U )
lcfrlem1.s	|- S = (Scalar ` U )
lcfrlem1.q	|- .X. = ( .r ` S )
lcfrlem1.z	|- .0. = ( 0g ` S )
lcfrlem1.i	|- I = ( invr ` S )
lcfrlem1.f	|- F = (LFnl ` U )
lcfrlem1.d	|- D = (LDual ` U )
lcfrlem1.t	|- .x. = ( .s ` D )
lcfrlem1.m	|- .- = ( -g ` D )
lcfrlem1.u	|- ( ph -> U e. LVec )
lcfrlem1.e	|- ( ph -> E e. F )
lcfrlem1.g	|- ( ph -> G e. F )
lcfrlem1.x	|- ( ph -> X e. V )
lcfrlem1.n	|- ( ph -> ( G ` X ) =/= .0. )
lcfrlem1.h	|- H = ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) )
lcfrlem2.l	|- L = (LKer ` U )

Assertion

Ref	Expression
lcfrlem2	|- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` H ) )

Proof of Theorem lcfrlem2

Step	Hyp	Ref	Expression
1		lcfrlem1.s	. . . . . 6 |- S = (Scalar ` U )
2		eqid 2453	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
3		lcfrlem1.f	. . . . . 6 |- F = (LFnl ` U )
4		lcfrlem2.l	. . . . . 6 |- L = (LKer ` U )
5		lcfrlem1.d	. . . . . 6 |- D = (LDual ` U )
6		lcfrlem1.t	. . . . . 6 |- .x. = ( .s ` D )
7		lcfrlem1.u	. . . . . 6 |- ( ph -> U e. LVec )
8		lcfrlem1.g	. . . . . 6 |- ( ph -> G e. F )
9		lveclmod 18341	. . . . . . . . 9 |- ( U e. LVec -> U e. LMod )
10	7, 9	syl 17	. . . . . . . 8 |- ( ph -> U e. LMod )
11	1	lmodring 18111	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
12	10, 11	syl 17	. . . . . . 7 |- ( ph -> S e. Ring )
13	1	lvecdrng 18340	. . . . . . . . 9 |- ( U e. LVec -> S e. DivRing )
14	7, 13	syl 17	. . . . . . . 8 |- ( ph -> S e. DivRing )
15		lcfrlem1.x	. . . . . . . . 9 |- ( ph -> X e. V )
16		lcfrlem1.v	. . . . . . . . . 10 |- V = ( Base ` U )
17	1, 2, 16, 3	lflcl 32642	. . . . . . . . 9 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
18	7, 8, 15, 17	syl3anc 1269	. . . . . . . 8 |- ( ph -> ( G ` X ) e. ( Base ` S ) )
19		lcfrlem1.n	. . . . . . . 8 |- ( ph -> ( G ` X ) =/= .0. )
20		lcfrlem1.z	. . . . . . . . 9 |- .0. = ( 0g ` S )
21		lcfrlem1.i	. . . . . . . . 9 |- I = ( invr ` S )
22	2, 20, 21	drnginvrcl 18004	. . . . . . . 8 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
23	14, 18, 19, 22	syl3anc 1269	. . . . . . 7 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
24		lcfrlem1.e	. . . . . . . 8 |- ( ph -> E e. F )
25	1, 2, 16, 3	lflcl 32642	. . . . . . . 8 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
26	7, 24, 15, 25	syl3anc 1269	. . . . . . 7 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
27		lcfrlem1.q	. . . . . . . 8 |- .X. = ( .r ` S )
28	2, 27	ringcl 17806	. . . . . . 7 |- ( ( S e. Ring /\ ( I ` ( G ` X ) ) e. ( Base ` S ) /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
29	12, 23, 26, 28	syl3anc 1269	. . . . . 6 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
30	1, 2, 3, 4, 5, 6, 7, 8, 29	lkrss 32746	. . . . 5 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) )
31	3, 1, 2, 5, 6, 10, 29, 8	ldualvscl 32717	. . . . . 6 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32		ringgrp 17797	. . . . . . . 8 |- ( S e. Ring -> S e. Grp )
33	12, 32	syl 17	. . . . . . 7 |- ( ph -> S e. Grp )
34		eqid 2453	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	2, 34	ringidcl 17813	. . . . . . . 8 |- ( S e. Ring -> ( 1r ` S ) e. ( Base ` S ) )
36	12, 35	syl 17	. . . . . . 7 |- ( ph -> ( 1r ` S ) e. ( Base ` S ) )
37		eqid 2453	. . . . . . . 8 |- ( invg ` S ) = ( invg ` S )
38	2, 37	grpinvcl 16723	. . . . . . 7 |- ( ( S e. Grp /\ ( 1r ` S ) e. ( Base ` S ) ) -> ( ( invg ` S ) ` ( 1r ` S ) ) e. ( Base ` S ) )
39	33, 36, 38	syl2anc 667	. . . . . 6 |- ( ph -> ( ( invg ` S ) ` ( 1r ` S ) ) e. ( Base ` S ) )
40	1, 2, 3, 4, 5, 6, 7, 31, 39	lkrss 32746	. . . . 5 |- ( ph -> ( L ` ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) C_ ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) )
41	30, 40	sstrd 3444	. . . 4 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) )
42		sslin 3660	. . . 4 |- ( ( L ` G ) C_ ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( ( L ` E ) i^i ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) ) )
43	41, 42	syl 17	. . 3 |- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( ( L ` E ) i^i ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) ) )
44		eqid 2453	. . . 4 |- ( +g ` D ) = ( +g ` D )
45	3, 1, 2, 5, 6, 10, 39, 31	ldualvscl 32717	. . . 4 |- ( ph -> ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) e. F )
46	3, 4, 5, 44, 10, 24, 45	lkrin 32742	. . 3 |- ( ph -> ( ( L ` E ) i^i ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) ) C_ ( L ` ( E ( +g ` D ) ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) ) )
47	43, 46	sstrd 3444	. 2 |- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` ( E ( +g ` D ) ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) ) )
48		lcfrlem1.h	. . . 4 |- H = ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) )
49	48	fveq2i 5873	. . 3 |- ( L ` H ) = ( L ` ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) )
50		lcfrlem1.m	. . . . 5 |- .- = ( -g ` D )
51	1, 37, 34, 3, 5, 44, 6, 50, 10, 24, 31	ldualvsub 32733	. . . 4 |- ( ph -> ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) = ( E ( +g ` D ) ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) )
52	51	fveq2d 5874	. . 3 |- ( ph -> ( L ` ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) = ( L ` ( E ( +g ` D ) ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) ) )
53	49, 52	syl5req 2500	. 2 |- ( ph -> ( L ` ( E ( +g ` D ) ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) ) = ( L ` H ) )
54	47, 53	sseqtrd 3470	1 |- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` H ) )

Syntax hints:   -> wi 4   = wceq 1446   e. wcel 1889   =/= wne 2624   i^i cin 3405   C_ wss 3406   ` cfv 5585  (class class class)co 6295   Basecbs 15133   +g cplusg 15202   .rcmulr 15203  Scalarcsca 15205   .scvsca 15206   0gc0g 15350   Grpcgrp 16681   invgcminusg 16682   -gcsg 16683   1rcur 17747   Ringcrg 17792   invrcinvr 17911   DivRingcdr 17987   LModclmod 18103   LVecclvec 18337  LFnlclfn 32635  LKerclk 32663  LDualcld 32701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-sca 15218  df-vsca 15219  df-0g 15352  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-grp 16685  df-minusg 16686  df-sbg 16687  df-cmn 17444  df-abl 17445  df-mgp 17736  df-ur 17748  df-ring 17794  df-oppr 17863  df-dvdsr 17881  df-unit 17882  df-invr 17912  df-drng 17989  df-lmod 18105  df-lss 18168  df-lvec 18338  df-lfl 32636  df-lkr 32664  df-ldual 32702

This theorem is referenced by:  lcfrlem35  35157