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Theorem lduallmodlem 29415
Description: Lemma for lduallmod 29416. (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 2285 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 lduallmod.w . . . 4  |-  ( ph  ->  W  e.  LMod )
51, 2, 3, 4ldualvbase 29389 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
65eqcomd 2290 . 2  |-  ( ph  ->  F  =  ( Base `  D ) )
7 eqidd 2286 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
8 lduallmod.r . . . 4  |-  R  =  (Scalar `  W )
9 lduallmod.o . . . 4  |-  O  =  (oppr
`  R )
10 eqid 2285 . . . 4  |-  (Scalar `  D )  =  (Scalar `  D )
118, 9, 2, 10, 4ldualsca 29395 . . 3  |-  ( ph  ->  (Scalar `  D )  =  O )
1211eqcomd 2290 . 2  |-  ( ph  ->  O  =  (Scalar `  D ) )
13 lduallmod.s . . 3  |-  .x.  =  ( .s `  D )
1413a1i 10 . 2  |-  ( ph  ->  .x.  =  ( .s
`  D ) )
15 lduallmod.k . . . 4  |-  K  =  ( Base `  R
)
169, 15opprbas 15413 . . 3  |-  K  =  ( Base `  O
)
1716a1i 10 . 2  |-  ( ph  ->  K  =  ( Base `  O ) )
18 eqid 2285 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
199, 18oppradd 15414 . . 3  |-  ( +g  `  R )  =  ( +g  `  O )
2019a1i 10 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  O ) )
2111fveq2d 5531 . 2  |-  ( ph  ->  ( .r `  (Scalar `  D ) )  =  ( .r `  O
) )
22 eqid 2285 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
239, 22oppr1 15418 . . 3  |-  ( 1r
`  R )  =  ( 1r `  O
)
2423a1i 10 . 2  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  O ) )
258lmodrng 15637 . . 3  |-  ( W  e.  LMod  ->  R  e. 
Ring )
269opprrng 15415 . . 3  |-  ( R  e.  Ring  ->  O  e. 
Ring )
274, 25, 263syl 18 . 2  |-  ( ph  ->  O  e.  Ring )
282, 4ldualgrp 29409 . 2  |-  ( ph  ->  D  e.  Grp )
2943ad2ant1 976 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  W  e.  LMod )
30 simp2 956 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  x  e.  K )
31 simp3 957 . . 3  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  y  e.  F )
321, 8, 15, 2, 13, 29, 30, 31ldualvscl 29402 . 2  |-  ( (
ph  /\  x  e.  K  /\  y  e.  F
)  ->  ( x  .x.  y )  e.  F
)
33 eqid 2285 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
344adantr 451 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  ->  W  e.  LMod )
35 simpr1 961 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  ->  x  e.  K )
36 simpr2 962 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  -> 
y  e.  F )
37 simpr3 963 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  F  /\  z  e.  F ) )  -> 
z  e.  F )
381, 8, 15, 2, 33, 13, 34, 35, 36, 37ldualvsdi1 29406 . 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 451 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  ->  W  e.  LMod )
40 simpr1 961 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  ->  x  e.  K )
41 simpr2 962 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  F ) )  -> 
y  e.  K )
42 simpr3 963 . . 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 29407 . 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 2285 . . 3  |-  ( .r
`  (Scalar `  D )
)  =  ( .r
`  (Scalar `  D )
)
451, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42ldualvsass2 29405 . 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 451 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  W  e.  LMod )
4915, 22rngidcl 15363 . . . . . 6  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  K )
504, 25, 493syl 18 . . . . 5  |-  ( ph  ->  ( 1r `  R
)  e.  K )
5150adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  ( 1r `  R )  e.  K )
52 simpr 447 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  x  e.  F )
531, 46, 8, 15, 47, 2, 13, 48, 51, 52ldualvs 29400 . . 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 29347 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
x  o F  .X.  ( V  X.  { ( 1r `  R ) } ) )  =  x )
5553, 54eqtrd 2317 . 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 15635 1  |-  ( ph  ->  D  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   {csn 3642    X. cxp 4689   ` cfv 5257  (class class class)co 5860    o Fcof 6078   Basecbs 13150   +g cplusg 13210   .rcmulr 13211  Scalarcsca 13213   .scvsca 13214   Ringcrg 15339   1rcur 15341  opprcoppr 15406   LModclmod 15629  LFnlclfn 29320  LDualcld 29386
This theorem is referenced by:  lduallmod  29416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-mnd 14369  df-grp 14491  df-minusg 14492  df-sbg 14493  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-lmod 15631  df-lfl 29321  df-ldual 29387
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