Theorem lcfrlem2 35497

Description: Lemma for lcfr 35539. (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 2451	. . . . . 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 17302	. . . . . . . . 9 |- ( U e. LVec -> U e. LMod )
10	7, 9	syl 16	. . . . . . . 8 |- ( ph -> U e. LMod )
11	1	lmodrng 17071	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
12	10, 11	syl 16	. . . . . . 7 |- ( ph -> S e. Ring )
13	1	lvecdrng 17301	. . . . . . . . 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 33018	. . . . . . . . 9 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
18	7, 8, 15, 17	syl3anc 1219	. . . . . . . 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 16964	. . . . . . . 8 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
23	14, 18, 19, 22	syl3anc 1219	. . . . . . 7 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
24		lcfrlem1.e	. . . . . . . 8 |- ( ph -> E e. F )
25	1, 2, 16, 3	lflcl 33018	. . . . . . . 8 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
26	7, 24, 15, 25	syl3anc 1219	. . . . . . 7 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
27		lcfrlem1.q	. . . . . . . 8 |- .X. = ( .r ` S )
28	2, 27	rngcl 16773	. . . . . . 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 1219	. . . . . 6 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
30	1, 2, 3, 4, 5, 6, 7, 8, 29	lkrss 33122	. . . . 5 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) )
31	3, 1, 2, 5, 6, 10, 29, 8	ldualvscl 33093	. . . . . 6 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32		rnggrp 16765	. . . . . . . 8 |- ( S e. Ring -> S e. Grp )
33	12, 32	syl 16	. . . . . . 7 |- ( ph -> S e. Grp )
34		eqid 2451	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	2, 34	rngidcl 16780	. . . . . . . 8 |- ( S e. Ring -> ( 1r ` S ) e. ( Base ` S ) )
36	12, 35	syl 16	. . . . . . 7 |- ( ph -> ( 1r ` S ) e. ( Base ` S ) )
37		eqid 2451	. . . . . . . 8 |- ( invg ` S ) = ( invg ` S )
38	2, 37	grpinvcl 15694	. . . . . . 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 33122	. . . . 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 3467	. . . 4 |- ( ph -> ( L ` G ) C_ ( L ` ( ( ( invg ` S ) ` ( 1r ` S ) ) .x. ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ) )
42		sslin 3677	. . . 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 2451	. . . 4 |- ( +g ` D ) = ( +g ` D )
45	3, 1, 2, 5, 6, 10, 39, 31	ldualvscl 33093	. . . 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 33118	. . 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 3467	. 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 5795	. . 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 33109	. . . 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 5796	. . 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 2505	. 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 3493	1 |- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` H ) )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   =/= wne 2644   i^i cin 3428   C_ wss 3429   ` cfv 5519  (class class class)co 6193   Basecbs 14285   +g cplusg 14349   .rcmulr 14350  Scalarcsca 14352   .scvsca 14353   0gc0g 14489   Grpcgrp 15521   invgcminusg 15522   -gcsg 15524   1rcur 16717   Ringcrg 16760   invrcinvr 16878   DivRingcdr 16947   LModclmod 17063   LVecclvec 17298  LFnlclfn 33011  LKerclk 33039  LDualcld 33077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-0g 14491  df-mnd 15526  df-grp 15656  df-minusg 15657  df-sbg 15658  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-drng 16949  df-lmod 17065  df-lss 17129  df-lvec 17299  df-lfl 33012  df-lkr 33040  df-ldual 33078

This theorem is referenced by:  lcfrlem35  35531