Theorem lcfrlem1 36831

Description: Lemma for lcfr 36874. 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 6192	. 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 2622	. . . 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 19106	. . . . 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 2622	. . . . 5 |- ( Base ` S ) = ( Base ` S )
14		lcfrlem1.t	. . . . 5 |- .x. = ( .s ` D )
15	4	lvecdrng 19105	. . . . . . . 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 34351	. . . . . . . 8 |- ( ( U e. LVec /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` S ) )
20	9, 17, 18, 19	syl3anc 1326	. . . . . . 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 18764	. . . . . . 7 |- ( ( S e. DivRing /\ ( G ` X ) e. ( Base ` S ) /\ ( G ` X ) =/= .0. ) -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
25	16, 20, 21, 24	syl3anc 1326	. . . . . 6 |- ( ph -> ( I ` ( G ` X ) ) e. ( Base ` S ) )
26	4, 13, 3, 6	lflcl 34351	. . . . . . 7 |- ( ( U e. LVec /\ E e. F /\ X e. V ) -> ( E ` X ) e. ( Base ` S ) )
27	9, 12, 18, 26	syl3anc 1326	. . . . . 6 |- ( ph -> ( E ` X ) e. ( Base ` S ) )
28		lcfrlem1.q	. . . . . . 7 |- .X. = ( .r ` S )
29	4, 13, 28	lmodmcl 18875	. . . . . 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 1326	. . . . 5 |- ( ph -> ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) e. ( Base ` S ) )
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 34426	. . . 4 |- ( ph -> ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) e. F )
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 34444	. . 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 34425	. . . . 5 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) )
34		eqid 2622	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
35	13, 22, 28, 34, 23	drnginvrr 18767	. . . . . . . 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 1326	. . . . . . 7 |- ( ph -> ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) = ( 1r ` S ) )
37	36	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( G ` X ) .X. ( I ` ( G ` X ) ) ) .X. ( E ` X ) ) = ( ( 1r ` S ) .X. ( E ` X ) ) )
38	4	lmodring 18871	. . . . . . . 8 |- ( U e. LMod -> S e. Ring )
39	11, 38	syl 17	. . . . . . 7 |- ( ph -> S e. Ring )
40	13, 28	ringass 18564	. . . . . . 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 1328	. . . . . 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 18571	. . . . . . 7 |- ( ( S e. Ring /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
43	39, 27, 42	syl2anc 693	. . . . . 6 |- ( ph -> ( ( 1r ` S ) .X. ( E ` X ) ) = ( E ` X ) )
44	37, 41, 43	3eqtr3d 2664	. . . . 5 |- ( ph -> ( ( G ` X ) .X. ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) ) = ( E ` X ) )
45	33, 44	eqtrd 2656	. . . 4 |- ( ph -> ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) = ( E ` X ) )
46	45	oveq2d 6666	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ` X ) ) = ( ( E ` X ) ( -g ` S ) ( E ` X ) ) )
47	4	lmodfgrp 18872	. . . . 5 |- ( U e. LMod -> S e. Grp )
48	11, 47	syl 17	. . . 4 |- ( ph -> S e. Grp )
49	13, 22, 5	grpsubid 17499	. . . 4 |- ( ( S e. Grp /\ ( E ` X ) e. ( Base ` S ) ) -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
50	48, 27, 49	syl2anc 693	. . 3 |- ( ph -> ( ( E ` X ) ( -g ` S ) ( E ` X ) ) = .0. )
51	32, 46, 50	3eqtrd 2660	. 2 |- ( ph -> ( ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) ) ` X ) = .0. )
52	2, 51	syl5eq 2668	1 |- ( ph -> ( H ` X ) = .0. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   1rcur 18501   Ringcrg 18547   invrcinvr 18671   DivRingcdr 18747   LModclmod 18863   LVecclvec 19102  LFnlclfn 34344  LDualcld 34410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lvec 19103  df-lfl 34345  df-ldual 34411

This theorem is referenced by:  lcfrlem3  36833