Theorem lcfrlem2 34544

Description: Lemma for lcfr 34586. (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 2402	. . . . . 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 17964	. . . . . . . . 9 |- ( U e. LVec -> U e. LMod )
10	7, 9	syl 17	. . . . . . . 8 |- ( ph -> U e. LMod )
11	1	lmodring 17732	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
12	10, 11	syl 17	. . . . . . 7 |- ( ph -> S e. Ring )
13	1	lvecdrng 17963	. . . . . . . . 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 32063	. . . . . . . . 9 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
18	7, 8, 15, 17	syl3anc 1230	. . . . . . . 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 17625	. . . . . . . 8 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
23	14, 18, 19, 22	syl3anc 1230	. . . . . . 7 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
24		lcfrlem1.e	. . . . . . . 8 |- ( ph -> E e. F )
25	1, 2, 16, 3	lflcl 32063	. . . . . . . 8 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
26	7, 24, 15, 25	syl3anc 1230	. . . . . . 7 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
27		lcfrlem1.q	. . . . . . . 8 |- .X. = ( .r ` S )
28	2, 27	ringcl 17424	. . . . . . 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 1230	. . . . . 6 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
30	1, 2, 3, 4, 5, 6, 7, 8, 29	lkrss 32167	. . . . 5 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) )
31	3, 1, 2, 5, 6, 10, 29, 8	ldualvscl 32138	. . . . . 6 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32		ringgrp 17415	. . . . . . . 8 |- ( S e. Ring -> S e. Grp )
33	12, 32	syl 17	. . . . . . 7 |- ( ph -> S e. Grp )
34		eqid 2402	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	2, 34	ringidcl 17431	. . . . . . . 8 |- ( S e. Ring -> ( 1r ` S ) e. ( Base ` S ) )
36	12, 35	syl 17	. . . . . . 7 |- ( ph -> ( 1r ` S ) e. ( Base ` S ) )
37		eqid 2402	. . . . . . . 8 |- ( invg ` S ) = ( invg ` S )
38	2, 37	grpinvcl 16311	. . . . . . 7 |- ( ( S e. Grp /\ ( 1r ` S ) e. ( Base ` S ) ) -> ( ( invg ` S ) ` ( 1r ` S ) ) e. ( Base ` S ) )
39	33, 36, 38	syl2anc 659	. . . . . 6 |- ( ph -> ( ( invg ` S ) ` ( 1r ` S ) ) e. ( Base ` S ) )
40	1, 2, 3, 4, 5, 6, 7, 31, 39	lkrss 32167	. . . . 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 3451	. . . 4 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) )
42		sslin 3664	. . . 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 2402	. . . 4 |- ( +g ` D ) = ( +g ` D )
45	3, 1, 2, 5, 6, 10, 39, 31	ldualvscl 32138	. . . 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 32163	. . 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 3451	. 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 5808	. . 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 32154	. . . 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 5809	. . 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 2456	. 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 3477	1 |- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` H ) )

Syntax hints:   -> wi 4   = wceq 1405   e. wcel 1842   =/= wne 2598   i^i cin 3412   C_ wss 3413   ` cfv 5525  (class class class)co 6234   Basecbs 14733   +g cplusg 14801   .rcmulr 14802  Scalarcsca 14804   .scvsca 14805   0gc0g 14946   Grpcgrp 16269   invgcminusg 16270   -gcsg 16271   1rcur 17365   Ringcrg 17410   invrcinvr 17532   DivRingcdr 17608   LModclmod 17724   LVecclvec 17960  LFnlclfn 32056  LKerclk 32084  LDualcld 32122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-om 6639  df-1st 6738  df-2nd 6739  df-tpos 6912  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-mulr 14815  df-sca 14817  df-vsca 14818  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-grp 16273  df-minusg 16274  df-sbg 16275  df-cmn 17016  df-abl 17017  df-mgp 17354  df-ur 17366  df-ring 17412  df-oppr 17484  df-dvdsr 17502  df-unit 17503  df-invr 17533  df-drng 17610  df-lmod 17726  df-lss 17791  df-lvec 17961  df-lfl 32057  df-lkr 32085  df-ldual 32123

This theorem is referenced by:  lcfrlem35  34578