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Theorem lduallmodlem 29887
Description: Lemma for lduallmod 29888. (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  |-  .+  =  o F ( +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.
StepHypRef Expression
1 lduallmod.f . . . 4  |-  F  =  (LFnl `  W )
2 lduallmod.d . . . 4  |-  D  =  (LDual `  W )
3 eqid 2435 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 lduallmod.w . . . 4  |-  ( ph  ->  W  e.  LMod )
51, 2, 3, 4ldualvbase 29861 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
65eqcomd 2440 . 2  |-  ( ph  ->  F  =  ( Base `  D ) )
7 eqidd 2436 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
8 lduallmod.r . . . 4  |-  R  =  (Scalar `  W )
9 lduallmod.o . . . 4  |-  O  =  (oppr
`  R )
10 eqid 2435 . . . 4  |-  (Scalar `  D )  =  (Scalar `  D )
118, 9, 2, 10, 4ldualsca 29867 . . 3  |-  ( ph  ->  (Scalar `  D )  =  O )
1211eqcomd 2440 . 2  |-  ( ph  ->  O  =  (Scalar `  D ) )
13 lduallmod.s . . 3  |-  .x.  =  ( .s `  D )
1413a1i 11 . 2  |-  ( ph  ->  .x.  =  ( .s
`  D ) )
15 lduallmod.k . . . 4  |-  K  =  ( Base `  R
)
169, 15opprbas 15726 . . 3  |-  K  =  ( Base `  O
)
1716a1i 11 . 2  |-  ( ph  ->  K  =  ( Base `  O ) )
18 eqid 2435 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
199, 18oppradd 15727 . . 3  |-  ( +g  `  R )  =  ( +g  `  O )
2019a1i 11 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  O ) )
2111fveq2d 5724 . 2  |-  ( ph  ->  ( .r `  (Scalar `  D ) )  =  ( .r `  O
) )
22 eqid 2435 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
239, 22oppr1 15731 . . 3  |-  ( 1r
`  R )  =  ( 1r `  O
)
2423a1i 11 . 2  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  O ) )
258lmodrng 15950 . . 3  |-  ( W  e.  LMod  ->  R  e. 
Ring )
269opprrng 15728 . . 3  |-  ( R  e.  Ring  ->  O  e. 
Ring )
274, 25, 263syl 19 . 2  |-  ( ph  ->  O  e.  Ring )
282, 4ldualgrp 29881 . 2  |-  ( ph  ->  D  e.  Grp )
2943ad2ant1 978 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  W  e.  LMod )
30 simp2 958 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  x  e.  K )
31 simp3 959 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  y  e.  F )
321, 8, 15, 2, 13, 29, 30, 31ldualvscl 29874 . 2  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  ( x  .x.  y )  e.  F
)
33 eqid 2435 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
344adantr 452 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  ->  W  e.  LMod )
35 simpr1 963 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  ->  x  e.  K )
36 simpr2 964 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  -> 
y  e.  F )
37 simpr3 965 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  -> 
z  e.  F )
381, 8, 15, 2, 33, 13, 34, 35, 36, 37ldualvsdi1 29878 . 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
) ) )
394adantr 452 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  ->  W  e.  LMod )
40 simpr1 963 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  ->  x  e.  K )
41 simpr2 964 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  -> 
y  e.  K )
42 simpr3 965 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  -> 
z  e.  F )
431, 8, 18, 15, 2, 33, 13, 39, 40, 41, 42ldualvsdi2 29879 . 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 2435 . . 3  |-  ( .r
`  (Scalar `  D )
)  =  ( .r
`  (Scalar `  D )
)
451, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42ldualvsass2 29877 . 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 )
484adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  W  e.  LMod )
4915, 22rngidcl 15676 . . . . . 6  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  K )
504, 25, 493syl 19 . . . . 5  |-  ( ph  ->  ( 1r `  R
)  e.  K )
5150adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  ( 1r `  R )  e.  K )
52 simpr 448 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  x  e.  F )
531, 46, 8, 15, 47, 2, 13, 48, 51, 52ldualvs 29872 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( 1r `  R
)  .x.  x )  =  ( x  o F  .X.  ( V  X.  { ( 1r `  R ) } ) ) )
5446, 8, 1, 15, 47, 22, 48, 52lfl1sc 29819 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
x  o F  .X.  ( V  X.  { ( 1r `  R ) } ) )  =  x )
5553, 54eqtrd 2467 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
( 1r `  R
)  .x.  x )  =  x )
566, 7, 12, 14, 17, 20, 21, 24, 27, 28, 32, 38, 43, 45, 55islmodd 15948 1  |-  ( ph  ->  D  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {csn 3806    X. cxp 4868   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525   Ringcrg 15652   1rcur 15654  opprcoppr 15719   LModclmod 15942  LFnlclfn 29792  LDualcld 29858
This theorem is referenced by:  lduallmod  29888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-lmod 15944  df-lfl 29793  df-ldual 29859
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