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Theorem lduallmodlem 33824
Description: Lemma for lduallmod 33825. (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.
StepHypRef Expression
1 lduallmod.f . . . 4  |-  F  =  (LFnl `  W )
2 lduallmod.d . . . 4  |-  D  =  (LDual `  W )
3 eqid 2460 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 lduallmod.w . . . 4  |-  ( ph  ->  W  e.  LMod )
51, 2, 3, 4ldualvbase 33798 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
65eqcomd 2468 . 2  |-  ( ph  ->  F  =  ( Base `  D ) )
7 eqidd 2461 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
8 lduallmod.r . . . 4  |-  R  =  (Scalar `  W )
9 lduallmod.o . . . 4  |-  O  =  (oppr
`  R )
10 eqid 2460 . . . 4  |-  (Scalar `  D )  =  (Scalar `  D )
118, 9, 2, 10, 4ldualsca 33804 . . 3  |-  ( ph  ->  (Scalar `  D )  =  O )
1211eqcomd 2468 . 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 17055 . . 3  |-  K  =  ( Base `  O
)
1716a1i 11 . 2  |-  ( ph  ->  K  =  ( Base `  O ) )
18 eqid 2460 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
199, 18oppradd 17056 . . 3  |-  ( +g  `  R )  =  ( +g  `  O )
2019a1i 11 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  O ) )
2111fveq2d 5861 . 2  |-  ( ph  ->  ( .r `  (Scalar `  D ) )  =  ( .r `  O
) )
22 eqid 2460 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
239, 22oppr1 17060 . . 3  |-  ( 1r
`  R )  =  ( 1r `  O
)
2423a1i 11 . 2  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  O ) )
258lmodrng 17296 . . 3  |-  ( W  e.  LMod  ->  R  e. 
Ring )
269opprrng 17057 . . 3  |-  ( R  e.  Ring  ->  O  e. 
Ring )
274, 25, 263syl 20 . 2  |-  ( ph  ->  O  e.  Ring )
282, 4ldualgrp 33818 . 2  |-  ( ph  ->  D  e.  Grp )
2943ad2ant1 1012 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  W  e.  LMod )
30 simp2 992 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  x  e.  K )
31 simp3 993 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  y  e.  F )
321, 8, 15, 2, 13, 29, 30, 31ldualvscl 33811 . 2  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  ( x  .x.  y )  e.  F
)
33 eqid 2460 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
344adantr 465 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  ->  W  e.  LMod )
35 simpr1 997 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  ->  x  e.  K )
36 simpr2 998 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  -> 
y  e.  F )
37 simpr3 999 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  -> 
z  e.  F )
381, 8, 15, 2, 33, 13, 34, 35, 36, 37ldualvsdi1 33815 . 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 465 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  ->  W  e.  LMod )
40 simpr1 997 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  ->  x  e.  K )
41 simpr2 998 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  -> 
y  e.  K )
42 simpr3 999 . . 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 33816 . 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 2460 . . 3  |-  ( .r
`  (Scalar `  D )
)  =  ( .r
`  (Scalar `  D )
)
451, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42ldualvsass2 33814 . 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 465 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  W  e.  LMod )
4915, 22rngidcl 16999 . . . . . 6  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  K )
504, 25, 493syl 20 . . . . 5  |-  ( ph  ->  ( 1r `  R
)  e.  K )
5150adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  ( 1r `  R )  e.  K )
52 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  x  e.  F )
531, 46, 8, 15, 47, 2, 13, 48, 51, 52ldualvs 33809 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( 1r `  R
)  .x.  x )  =  ( x  oF  .X.  ( V  X.  { ( 1r `  R ) } ) ) )
5446, 8, 1, 15, 47, 22, 48, 52lfl1sc 33756 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
x  oF  .X.  ( V  X.  { ( 1r `  R ) } ) )  =  x )
5553, 54eqtrd 2501 . 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 17294 1  |-  ( ph  ->  D  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {csn 4020    X. cxp 4990   ` cfv 5579  (class class class)co 6275    oFcof 6513   Basecbs 14479   +g cplusg 14544   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548   1rcur 16936   Ringcrg 16979  opprcoppr 17048   LModclmod 17288  LFnlclfn 33729  LDualcld 33795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-lmod 17290  df-lfl 33730  df-ldual 33796
This theorem is referenced by:  lduallmod  33825
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