Theorem lcfrlem1 34526

Description: Lemma for lcfr 34569. 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 5804	. 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 2400	. . . 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 17962	. . . . 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 2400	. . . . 5 |- ( Base ` S ) = ( Base ` S )
14		lcfrlem1.t	. . . . 5 |- .x. = ( .s ` D )
15	4	lvecdrng 17961	. . . . . . . 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 32046	. . . . . . . 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 17623	. . . . . . 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 32046	. . . . . . 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 17734	. . . . . 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 32121	. . . 4 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 32139	. . 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 32120	. . . . 5 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
34		eqid 2400	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	13, 22, 28, 34, 23	drnginvrr 17626	. . . . . . . 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 6247	. . . . . 6 |- ( ph -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( 1r ` S ) .X. ( E ` X ) ) )
38	4	lmodring 17730	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
39	11, 38	syl 17	. . . . . . 7 |- ( ph -> S e. Ring )
40	13, 28	ringass 17425	. . . . . . 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 17432	. . . . . . 7 |- ( ( S e. Ring /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
43	39, 27, 42	syl2anc 659	. . . . . 6 |- ( ph -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
44	37, 41, 43	3eqtr3d 2449	. . . . 5 |- ( ph -> ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) = ( E ` X ) )
45	33, 44	eqtrd 2441	. . . 4 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( E ` X ) )
46	45	oveq2d 6248	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) ) = ( ( E ` X ) ( -g ` S ) ( E ` X ) ) )
47	4	lmodfgrp 17731	. . . . 5 |- ( U e. LMod -> S e. Grp )
48	11, 47	syl 17	. . . 4 |- ( ph -> S e. Grp )
49	13, 22, 5	grpsubid 16336	. . . 4 |- ( ( S e. Grp /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
50	48, 27, 49	syl2anc 659	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
51	32, 46, 50	3eqtrd 2445	. 2 |- ( ph -> ( ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ` X ) = .0. )
52	2, 51	syl5eq 2453	1 |- ( ph -> ( H ` X ) = .0. )

Syntax hints:   -> wi 4   = wceq 1403   e. wcel 1840   =/= wne 2596   ` cfv 5523  (class class class)co 6232   Basecbs 14731   .rcmulr 14800  Scalarcsca 14802   .scvsca 14803   0gc0g 14944   Grpcgrp 16267   -gcsg 16269   1rcur 17363   Ringcrg 17408   invrcinvr 17530   DivRingcdr 17606   LModclmod 17722   LVecclvec 17958  LFnlclfn 32039  LDualcld 32105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-tpos 6910  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-sca 14815  df-vsca 14816  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-minusg 16272  df-sbg 16273  df-cmn 17014  df-abl 17015  df-mgp 17352  df-ur 17364  df-ring 17410  df-oppr 17482  df-dvdsr 17500  df-unit 17501  df-invr 17531  df-drng 17608  df-lmod 17724  df-lvec 17959  df-lfl 32040  df-ldual 32106

This theorem is referenced by:  lcfrlem3  34528