Theorem lcfrlem2 36340

Description: Lemma for lcfr 36382. (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 2467	. . . . . 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 17535	. . . . . . . . 9 |- ( U e. LVec -> U e. LMod )
10	7, 9	syl 16	. . . . . . . 8 |- ( ph -> U e. LMod )
11	1	lmodrng 17303	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
12	10, 11	syl 16	. . . . . . 7 |- ( ph -> S e. Ring )
13	1	lvecdrng 17534	. . . . . . . . 9 |- ( U e. LVec -> S e. DivRing )
14	7, 13	syl 16	. . . . . . . 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 33861	. . . . . . . . 9 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
18	7, 8, 15, 17	syl3anc 1228	. . . . . . . 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 17196	. . . . . . . 8 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
23	14, 18, 19, 22	syl3anc 1228	. . . . . . 7 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
24		lcfrlem1.e	. . . . . . . 8 |- ( ph -> E e. F )
25	1, 2, 16, 3	lflcl 33861	. . . . . . . 8 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
26	7, 24, 15, 25	syl3anc 1228	. . . . . . 7 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
27		lcfrlem1.q	. . . . . . . 8 |- .X. = ( .r ` S )
28	2, 27	rngcl 16999	. . . . . . 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 1228	. . . . . 6 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
30	1, 2, 3, 4, 5, 6, 7, 8, 29	lkrss 33965	. . . . 5 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) )
31	3, 1, 2, 5, 6, 10, 29, 8	ldualvscl 33936	. . . . . 6 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32		rnggrp 16991	. . . . . . . 8 |- ( S e. Ring -> S e. Grp )
33	12, 32	syl 16	. . . . . . 7 |- ( ph -> S e. Grp )
34		eqid 2467	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	2, 34	rngidcl 17006	. . . . . . . 8 |- ( S e. Ring -> ( 1r ` S ) e. ( Base ` S ) )
36	12, 35	syl 16	. . . . . . 7 |- ( ph -> ( 1r ` S ) e. ( Base ` S ) )
37		eqid 2467	. . . . . . . 8 |- ( invg ` S ) = ( invg ` S )
38	2, 37	grpinvcl 15896	. . . . . . 7 |- ( ( S e. Grp /\ ( 1r ` S ) e. ( Base ` S ) ) -> ( ( invg ` S ) ` ( 1r ` S ) ) e. ( Base ` S ) )
39	33, 36, 38	syl2anc 661	. . . . . 6 |- ( ph -> ( ( invg ` S ) ` ( 1r ` S ) ) e. ( Base ` S ) )
40	1, 2, 3, 4, 5, 6, 7, 31, 39	lkrss 33965	. . . . 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 3514	. . . 4 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) )
42		sslin 3724	. . . 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 16	. . 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 2467	. . . 4 |- ( +g ` D ) = ( +g ` D )
45	3, 1, 2, 5, 6, 10, 39, 31	ldualvscl 33936	. . . 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 33961	. . 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 3514	. 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 5867	. . 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 33952	. . . 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 5868	. . 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 2521	. 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 3540	1 |- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` H ) )

Syntax hints:   -> wi 4   = wceq 1379   e. wcel 1767   =/= wne 2662   i^i cin 3475   C_ wss 3476   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   .rcmulr 14552  Scalarcsca 14554   .scvsca 14555   0gc0g 14691   Grpcgrp 15723   invgcminusg 15724   -gcsg 15726   1rcur 16943   Ringcrg 16986   invrcinvr 17104   DivRingcdr 17179   LModclmod 17295   LVecclvec 17531  LFnlclfn 33854  LKerclk 33882  LDualcld 33920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-sbg 15860  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-drng 17181  df-lmod 17297  df-lss 17362  df-lvec 17532  df-lfl 33855  df-lkr 33883  df-ldual 33921

This theorem is referenced by:  lcfrlem35  36374