Theorem lcfrlem1 36740

Description: Lemma for lcfr 36783. 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 5873	. 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 2467	. . . 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 17623	. . . . 5 |- ( U e. LVec -> U e. LMod )
11	9, 10	syl 16	. . . 4 |- ( ph -> U e. LMod )
12		lcfrlem1.e	. . . 4 |- ( ph -> E e. F )
13		eqid 2467	. . . . 5 |- ( Base ` S ) = ( Base ` S )
14		lcfrlem1.t	. . . . 5 |- .x. = ( .s ` D )
15	4	lvecdrng 17622	. . . . . . . 8 |- ( U e. LVec -> S e. DivRing )
16	9, 15	syl 16	. . . . . . 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 34262	. . . . . . . 8 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
20	9, 17, 18, 19	syl3anc 1228	. . . . . . 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 17284	. . . . . . 7 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
25	16, 20, 21, 24	syl3anc 1228	. . . . . 6 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
26	4, 13, 3, 6	lflcl 34262	. . . . . . 7 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
27	9, 12, 18, 26	syl3anc 1228	. . . . . 6 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
28		lcfrlem1.q	. . . . . . 7 |- .X. = ( .r ` S )
29	4, 13, 28	lmodmcl 17395	. . . . . 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 1228	. . . . 5 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 34337	. . . 4 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 34355	. . 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 34336	. . . . 5 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
34		eqid 2467	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	13, 22, 28, 34, 23	drnginvrr 17287	. . . . . . . 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 1228	. . . . . . 7 |- ( ph -> ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) = ( 1r ` S ) )
37	36	oveq1d 6310	. . . . . 6 |- ( ph -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( 1r ` S ) .X. ( E ` X ) ) )
38	4	lmodring 17391	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
39	11, 38	syl 16	. . . . . . 7 |- ( ph -> S e. Ring )
40	13, 28	ringass 17087	. . . . . . 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 1230	. . . . . 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 17094	. . . . . . 7 |- ( ( S e. Ring /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
43	39, 27, 42	syl2anc 661	. . . . . 6 |- ( ph -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
44	37, 41, 43	3eqtr3d 2516	. . . . 5 |- ( ph -> ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) = ( E ` X ) )
45	33, 44	eqtrd 2508	. . . 4 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( E ` X ) )
46	45	oveq2d 6311	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) ) = ( ( E ` X ) ( -g ` S ) ( E ` X ) ) )
47	4	lmodfgrp 17392	. . . . 5 |- ( U e. LMod -> S e. Grp )
48	11, 47	syl 16	. . . 4 |- ( ph -> S e. Grp )
49	13, 22, 5	grpsubid 15994	. . . 4 |- ( ( S e. Grp /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
50	48, 27, 49	syl2anc 661	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
51	32, 46, 50	3eqtrd 2512	. 2 |- ( ph -> ( ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ` X ) = .0. )
52	2, 51	syl5eq 2520	1 |- ( ph -> ( H ` X ) = .0. )

Syntax hints:   -> wi 4   = wceq 1379   e. wcel 1767   =/= wne 2662   ` cfv 5594  (class class class)co 6295   Basecbs 14507   .rcmulr 14573  Scalarcsca 14575   .scvsca 14576   0gc0g 14712   Grpcgrp 15925   -gcsg 15927   1rcur 17025   Ringcrg 17070   invrcinvr 17192   DivRingcdr 17267   LModclmod 17383   LVecclvec 17619  LFnlclfn 34255  LDualcld 34321

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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-lmod 17385  df-lvec 17620  df-lfl 34256  df-ldual 34322

This theorem is referenced by:  lcfrlem3  36742