Theorem lcfrlem1 35192

Description: Lemma for lcfr 35235. 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 5697	. 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 2443	. . . 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 17192	. . . . 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 2443	. . . . 5 |- ( Base ` S ) = ( Base ` S )
14		lcfrlem1.t	. . . . 5 |- .x. = ( .s ` D )
15	4	lvecdrng 17191	. . . . . . . 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 32714	. . . . . . . 8 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
20	9, 17, 18, 19	syl3anc 1218	. . . . . . 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 16854	. . . . . . 7 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
25	16, 20, 21, 24	syl3anc 1218	. . . . . 6 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
26	4, 13, 3, 6	lflcl 32714	. . . . . . 7 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
27	9, 12, 18, 26	syl3anc 1218	. . . . . 6 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
28		lcfrlem1.q	. . . . . . 7 |- .X. = ( .r ` S )
29	4, 13, 28	lmodmcl 16965	. . . . . 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 1218	. . . . 5 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 32789	. . . 4 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 32807	. . 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 32788	. . . . 5 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
34		eqid 2443	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	13, 22, 28, 34, 23	drnginvrr 16857	. . . . . . . 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 1218	. . . . . . 7 |- ( ph -> ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) = ( 1r ` S ) )
37	36	oveq1d 6111	. . . . . 6 |- ( ph -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( 1r ` S ) .X. ( E ` X ) ) )
38	4	lmodrng 16961	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
39	11, 38	syl 16	. . . . . . 7 |- ( ph -> S e. Ring )
40	13, 28	rngass 16666	. . . . . . 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 1220	. . . . . 6 |- ( ph -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
42	13, 28, 34	rnglidm 16673	. . . . . . 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 2483	. . . . 5 |- ( ph -> ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) = ( E ` X ) )
45	33, 44	eqtrd 2475	. . . 4 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( E ` X ) )
46	45	oveq2d 6112	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) ) = ( ( E ` X ) ( -g ` S ) ( E ` X ) ) )
47	4	lmodfgrp 16962	. . . . 5 |- ( U e. LMod -> S e. Grp )
48	11, 47	syl 16	. . . 4 |- ( ph -> S e. Grp )
49	13, 22, 5	grpsubid 15615	. . . 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 2479	. 2 |- ( ph -> ( ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ` X ) = .0. )
52	2, 51	syl5eq 2487	1 |- ( ph -> ( H ` X ) = .0. )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   =/= wne 2611   ` cfv 5423  (class class class)co 6096   Basecbs 14179   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247   0gc0g 14383   Grpcgrp 15415   -gcsg 15418   1rcur 16608   Ringcrg 16650   invrcinvr 16768   DivRingcdr 16837   LModclmod 16953   LVecclvec 17188  LFnlclfn 32707  LDualcld 32773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-drng 16839  df-lmod 16955  df-lvec 17189  df-lfl 32708  df-ldual 32774

This theorem is referenced by:  lcfrlem3  35194