Theorem lduallmodlem 34439

Description: Lemma for lduallmod 34440. (Contributed by NM, 22-Oct-2014.)

Hypotheses

Ref	Expression
lduallmod.d	|- D = (LDual ` W )
lduallmod.w	|- ( ph -> W e. LMod )
lduallmod.v	|- V = ( Base ` W )
lduallmod.p	|- .+ = oF ( +g ` W )
lduallmod.f	|- F = (LFnl ` W )
lduallmod.r	|- R = (Scalar ` W )
lduallmod.k	|- K = ( Base ` R )
lduallmod.t	|- .X. = ( .r ` R )
lduallmod.o	|- O = (oppr ` R )
lduallmod.s	|- .x. = ( .s ` D )

Assertion

Ref	Expression
lduallmodlem	|- ( ph -> D e. LMod )

Proof of Theorem lduallmodlem

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lduallmod.f	. . . 4 |- F = (LFnl ` W )
2		lduallmod.d	. . . 4 |- D = (LDual ` W )
3		eqid 2622	. . . 4 |- ( Base ` D ) = ( Base ` D )
4		lduallmod.w	. . . 4 |- ( ph -> W e. LMod )
5	1, 2, 3, 4	ldualvbase 34413	. . 3 |- ( ph -> ( Base ` D ) = F )
6	5	eqcomd 2628	. 2 |- ( ph -> F = ( Base ` D ) )
7		eqidd 2623	. 2 |- ( ph -> ( +g ` D ) = ( +g ` D ) )
8		lduallmod.r	. . . 4 |- R = (Scalar ` W )
9		lduallmod.o	. . . 4 |- O = (oppr ` R )
10		eqid 2622	. . . 4 |- (Scalar ` D ) = (Scalar ` D )
11	8, 9, 2, 10, 4	ldualsca 34419	. . 3 |- ( ph -> (Scalar ` D ) = O )
12	11	eqcomd 2628	. 2 |- ( ph -> O = (Scalar ` D ) )
13		lduallmod.s	. . 3 |- .x. = ( .s ` D )
14	13	a1i 11	. 2 |- ( ph -> .x. = ( .s ` D ) )
15		lduallmod.k	. . . 4 |- K = ( Base ` R )
16	9, 15	opprbas 18629	. . 3 |- K = ( Base ` O )
17	16	a1i 11	. 2 |- ( ph -> K = ( Base ` O ) )
18		eqid 2622	. . . 4 |- ( +g ` R ) = ( +g ` R )
19	9, 18	oppradd 18630	. . 3 |- ( +g ` R ) = ( +g ` O )
20	19	a1i 11	. 2 |- ( ph -> ( +g ` R ) = ( +g ` O ) )
21	11	fveq2d 6195	. 2 |- ( ph -> ( .r ` (Scalar ` D ) ) = ( .r ` O ) )
22		eqid 2622	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
23	9, 22	oppr1 18634	. . 3 |- ( 1r ` R ) = ( 1r ` O )
24	23	a1i 11	. 2 |- ( ph -> ( 1r ` R ) = ( 1r ` O ) )
25	8	lmodring 18871	. . 3 |- ( W e. LMod -> R e. Ring )
26	9	opprring 18631	. . 3 |- ( R e. Ring -> O e. Ring )
27	4, 25, 26	3syl 18	. 2 |- ( ph -> O e. Ring )
28	2, 4	ldualgrp 34433	. 2 |- ( ph -> D e. Grp )
29	4	3ad2ant1 1082	. . 3 |- ( ( ph /\ x e. K /\ y e. F ) -> W e. LMod )
30		simp2 1062	. . 3 |- ( ( ph /\ x e. K /\ y e. F ) -> x e. K )
31		simp3 1063	. . 3 |- ( ( ph /\ x e. K /\ y e. F ) -> y e. F )
32	1, 8, 15, 2, 13, 29, 30, 31	ldualvscl 34426	. 2 |- ( ( ph /\ x e. K /\ y e. F ) -> ( x .x. y ) e. F )
33		eqid 2622	. . 3 |- ( +g ` D ) = ( +g ` D )
34	4	adantr 481	. . 3 |- ( ( ph /\ ( x e. K /\ y e. F /\ z e. F ) ) -> W e. LMod )
35		simpr1 1067	. . 3 |- ( ( ph /\ ( x e. K /\ y e. F /\ z e. F ) ) -> x e. K )
36		simpr2 1068	. . 3 |- ( ( ph /\ ( x e. K /\ y e. F /\ z e. F ) ) -> y e. F )
37		simpr3 1069	. . 3 |- ( ( ph /\ ( x e. K /\ y e. F /\ z e. F ) ) -> z e. F )
38	1, 8, 15, 2, 33, 13, 34, 35, 36, 37	ldualvsdi1 34430	. 2 |- ( ( ph /\ ( x e. K /\ y e. F /\ z e. F ) ) -> ( x .x. ( y ( +g ` D ) z ) ) = ( ( x .x. y ) ( +g ` D ) ( x .x. z ) ) )
39	4	adantr 481	. . 3 |- ( ( ph /\ ( x e. K /\ y e. K /\ z e. F ) ) -> W e. LMod )
40		simpr1 1067	. . 3 |- ( ( ph /\ ( x e. K /\ y e. K /\ z e. F ) ) -> x e. K )
41		simpr2 1068	. . 3 |- ( ( ph /\ ( x e. K /\ y e. K /\ z e. F ) ) -> y e. K )
42		simpr3 1069	. . 3 |- ( ( ph /\ ( x e. K /\ y e. K /\ z e. F ) ) -> z e. F )
43	1, 8, 18, 15, 2, 33, 13, 39, 40, 41, 42	ldualvsdi2 34431	. 2 |- ( ( ph /\ ( x e. K /\ y e. K /\ z e. F ) ) -> ( ( x ( +g ` R ) y ) .x. z ) = ( ( x .x. z ) ( +g ` D ) ( y .x. z ) ) )
44		eqid 2622	. . 3 |- ( .r ` (Scalar ` D ) ) = ( .r ` (Scalar ` D ) )
45	1, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42	ldualvsass2 34429	. 2 |- ( ( ph /\ ( x e. K /\ y e. K /\ z e. F ) ) -> ( ( x ( .r ` (Scalar ` D ) ) y ) .x. z ) = ( x .x. ( y .x. z ) ) )
46		lduallmod.v	. . . 4 |- V = ( Base ` W )
47		lduallmod.t	. . . 4 |- .X. = ( .r ` R )
48	4	adantr 481	. . . 4 |- ( ( ph /\ x e. F ) -> W e. LMod )
49	15, 22	ringidcl 18568	. . . . . 6 |- ( R e. Ring -> ( 1r ` R ) e. K )
50	4, 25, 49	3syl 18	. . . . 5 |- ( ph -> ( 1r ` R ) e. K )
51	50	adantr 481	. . . 4 |- ( ( ph /\ x e. F ) -> ( 1r ` R ) e. K )
52		simpr 477	. . . 4 |- ( ( ph /\ x e. F ) -> x e. F )
53	1, 46, 8, 15, 47, 2, 13, 48, 51, 52	ldualvs 34424	. . 3 |- ( ( ph /\ x e. F ) -> ( ( 1r ` R ) .x. x ) = ( x oF .X. ( V X. { ( 1r ` R ) } ) ) )
54	46, 8, 1, 15, 47, 22, 48, 52	lfl1sc 34371	. . 3 |- ( ( ph /\ x e. F ) -> ( x oF .X. ( V X. { ( 1r ` R ) } ) ) = x )
55	53, 54	eqtrd 2656	. 2 |- ( ( ph /\ x e. F ) -> ( ( 1r ` R ) .x. x ) = x )
56	6, 7, 12, 14, 17, 20, 21, 24, 27, 28, 32, 38, 43, 45, 55	islmodd 18869	1 |- ( ph -> D e. LMod )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622   LModclmod 18863  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-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-lmod 18865  df-lfl 34345  df-ldual 34411

This theorem is referenced by:  lduallmod  34440