Theorem lcfrlem1 35156

Description: Lemma for lcfr 35199. Note that X is z in Mario's notes. (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 ) )

Assertion

Ref	Expression
lcfrlem1	|- ( ph -> ( H ` X ) = .0. )

Proof of Theorem lcfrlem1

Step	Hyp	Ref	Expression
1		lcfrlem1.h	. . 3 |- H = ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) )
2	1	fveq1i 5893	. 2 |- ( H ` X ) = ( ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ` X )
3		lcfrlem1.v	. . . 4 |- V = ( Base ` U )
4		lcfrlem1.s	. . . 4 |- S = (Scalar ` U )
5		eqid 2462	. . . 4 |- ( -g ` S ) = ( -g ` S )
6		lcfrlem1.f	. . . 4 |- F = (LFnl ` U )
7		lcfrlem1.d	. . . 4 |- D = (LDual ` U )
8		lcfrlem1.m	. . . 4 |- .- = ( -g ` D )
9		lcfrlem1.u	. . . . 5 |- ( ph -> U e. LVec )
10		lveclmod 18384	. . . . 5 |- ( U e. LVec -> U e. LMod )
11	9, 10	syl 17	. . . 4 |- ( ph -> U e. LMod )
12		lcfrlem1.e	. . . 4 |- ( ph -> E e. F )
13		eqid 2462	. . . . 5 |- ( Base ` S ) = ( Base ` S )
14		lcfrlem1.t	. . . . 5 |- .x. = ( .s ` D )
15	4	lvecdrng 18383	. . . . . . . 8 |- ( U e. LVec -> S e. DivRing )
16	9, 15	syl 17	. . . . . . 7 |- ( ph -> S e. DivRing )
17		lcfrlem1.g	. . . . . . . 8 |- ( ph -> G e. F )
18		lcfrlem1.x	. . . . . . . 8 |- ( ph -> X e. V )
19	4, 13, 3, 6	lflcl 32676	. . . . . . . 8 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
20	9, 17, 18, 19	syl3anc 1276	. . . . . . 7 |- ( ph -> ( G ` X ) e. ( Base ` S ) )
21		lcfrlem1.n	. . . . . . 7 |- ( ph -> ( G ` X ) =/= .0. )
22		lcfrlem1.z	. . . . . . . 8 |- .0. = ( 0g ` S )
23		lcfrlem1.i	. . . . . . . 8 |- I = ( invr ` S )
24	13, 22, 23	drnginvrcl 18047	. . . . . . 7 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
25	16, 20, 21, 24	syl3anc 1276	. . . . . 6 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
26	4, 13, 3, 6	lflcl 32676	. . . . . . 7 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
27	9, 12, 18, 26	syl3anc 1276	. . . . . 6 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
28		lcfrlem1.q	. . . . . . 7 |- .X. = ( .r ` S )
29	4, 13, 28	lmodmcl 18158	. . . . . 6 |- ( ( U e. LMod /\ ( I ` ( G ` X ) ) e. ( Base ` S ) /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
30	11, 25, 27, 29	syl3anc 1276	. . . . 5 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 32751	. . . 4 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 32769	. . 3 |- ( ph -> ( ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ` X ) = ( ( E ` X ) ( -g ` S ) ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) ) )
33	6, 3, 4, 13, 28, 7, 14, 9, 30, 17, 18	ldualvsval 32750	. . . . 5 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
34		eqid 2462	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	13, 22, 28, 34, 23	drnginvrr 18050	. . . . . . . 8 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) = ( 1r ` S ) )
36	16, 20, 21, 35	syl3anc 1276	. . . . . . 7 |- ( ph -> ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) = ( 1r ` S ) )
37	36	oveq1d 6335	. . . . . 6 |- ( ph -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( 1r ` S ) .X. ( E ` X ) ) )
38	4	lmodring 18154	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
39	11, 38	syl 17	. . . . . . 7 |- ( ph -> S e. Ring )
40	13, 28	ringass 17852	. . . . . . 7 |- ( ( S e. Ring /\ ( ( G ` X ) e. ( Base ` S ) /\ ( I ` ( G ` X ) ) e. ( Base ` S ) /\ ( E ` X ) e. ( Base ` S ) ) ) -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
41	39, 20, 25, 27, 40	syl13anc 1278	. . . . . 6 |- ( ph -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
42	13, 28, 34	ringlidm 17859	. . . . . . 7 |- ( ( S e. Ring /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
43	39, 27, 42	syl2anc 671	. . . . . 6 |- ( ph -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
44	37, 41, 43	3eqtr3d 2504	. . . . 5 |- ( ph -> ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) = ( E ` X ) )
45	33, 44	eqtrd 2496	. . . 4 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( E ` X ) )
46	45	oveq2d 6336	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) ) = ( ( E ` X ) ( -g ` S ) ( E ` X ) ) )
47	4	lmodfgrp 18155	. . . . 5 |- ( U e. LMod -> S e. Grp )
48	11, 47	syl 17	. . . 4 |- ( ph -> S e. Grp )
49	13, 22, 5	grpsubid 16793	. . . 4 |- ( ( S e. Grp /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
50	48, 27, 49	syl2anc 671	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
51	32, 46, 50	3eqtrd 2500	. 2 |- ( ph -> ( ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ` X ) = .0. )
52	2, 51	syl5eq 2508	1 |- ( ph -> ( H ` X ) = .0. )

Syntax hints:   -> wi 4   = wceq 1455   e. wcel 1898   =/= wne 2633   ` cfv 5605  (class class class)co 6320   Basecbs 15176   .rcmulr 15246  Scalarcsca 15248   .scvsca 15249   0gc0g 15393   Grpcgrp 16724   -gcsg 16726   1rcur 17790   Ringcrg 17835   invrcinvr 17954   DivRingcdr 18030   LModclmod 18146   LVecclvec 18380  LFnlclfn 32669  LDualcld 32735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-of 6563  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-struct 15178  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-sca 15261  df-vsca 15262  df-0g 15395  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-grp 16728  df-minusg 16729  df-sbg 16730  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-drng 18032  df-lmod 18148  df-lvec 18381  df-lfl 32670  df-ldual 32736

This theorem is referenced by:  lcfrlem3  35158