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Theorem frlmup2 27251
Description: The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Hypotheses
Ref Expression
frlmup.f  |-  F  =  ( R freeLMod  I )
frlmup.b  |-  B  =  ( Base `  F
)
frlmup.c  |-  C  =  ( Base `  T
)
frlmup.v  |-  .x.  =  ( .s `  T )
frlmup.e  |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F 
.x.  A ) ) )
frlmup.t  |-  ( ph  ->  T  e.  LMod )
frlmup.i  |-  ( ph  ->  I  e.  X )
frlmup.r  |-  ( ph  ->  R  =  (Scalar `  T ) )
frlmup.a  |-  ( ph  ->  A : I --> C )
frlmup.y  |-  ( ph  ->  Y  e.  I )
frlmup.u  |-  U  =  ( R unitVec  I )
Assertion
Ref Expression
frlmup2  |-  ( ph  ->  ( E `  ( U `  Y )
)  =  ( A `
 Y ) )
Distinct variable groups:    x, R    x, I    x, F    x, B    x, C    x,  .x.    x, A    x, X    ph, x    x, Y    x, U    x, T
Allowed substitution hint:    E( x)

Proof of Theorem frlmup2
StepHypRef Expression
1 frlmup.r . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  T ) )
2 frlmup.t . . . . . . 7  |-  ( ph  ->  T  e.  LMod )
3 eqid 2283 . . . . . . . 8  |-  (Scalar `  T )  =  (Scalar `  T )
43lmodrng 15635 . . . . . . 7  |-  ( T  e.  LMod  ->  (Scalar `  T )  e.  Ring )
52, 4syl 15 . . . . . 6  |-  ( ph  ->  (Scalar `  T )  e.  Ring )
61, 5eqeltrd 2357 . . . . 5  |-  ( ph  ->  R  e.  Ring )
7 frlmup.i . . . . 5  |-  ( ph  ->  I  e.  X )
8 frlmup.u . . . . . 6  |-  U  =  ( R unitVec  I )
9 frlmup.f . . . . . 6  |-  F  =  ( R freeLMod  I )
10 frlmup.b . . . . . 6  |-  B  =  ( Base `  F
)
118, 9, 10uvcff 27240 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  U : I --> B )
126, 7, 11syl2anc 642 . . . 4  |-  ( ph  ->  U : I --> B )
13 frlmup.y . . . 4  |-  ( ph  ->  Y  e.  I )
14 ffvelrn 5663 . . . 4  |-  ( ( U : I --> B  /\  Y  e.  I )  ->  ( U `  Y
)  e.  B )
1512, 13, 14syl2anc 642 . . 3  |-  ( ph  ->  ( U `  Y
)  e.  B )
16 oveq1 5865 . . . . 5  |-  ( x  =  ( U `  Y )  ->  (
x  o F  .x.  A )  =  ( ( U `  Y
)  o F  .x.  A ) )
1716oveq2d 5874 . . . 4  |-  ( x  =  ( U `  Y )  ->  ( T  gsumg  ( x  o F 
.x.  A ) )  =  ( T  gsumg  ( ( U `  Y )  o F  .x.  A
) ) )
18 frlmup.e . . . 4  |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F 
.x.  A ) ) )
19 ovex 5883 . . . 4  |-  ( T 
gsumg  ( ( U `  Y )  o F 
.x.  A ) )  e.  _V
2017, 18, 19fvmpt 5602 . . 3  |-  ( ( U `  Y )  e.  B  ->  ( E `  ( U `  Y ) )  =  ( T  gsumg  ( ( U `  Y )  o F 
.x.  A ) ) )
2115, 20syl 15 . 2  |-  ( ph  ->  ( E `  ( U `  Y )
)  =  ( T 
gsumg  ( ( U `  Y )  o F 
.x.  A ) ) )
22 frlmup.c . . 3  |-  C  =  ( Base `  T
)
23 eqid 2283 . . 3  |-  ( 0g
`  T )  =  ( 0g `  T
)
24 lmodcmn 15673 . . . 4  |-  ( T  e.  LMod  ->  T  e. CMnd
)
25 cmnmnd 15104 . . . 4  |-  ( T  e. CMnd  ->  T  e.  Mnd )
262, 24, 253syl 18 . . 3  |-  ( ph  ->  T  e.  Mnd )
27 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
28 frlmup.v . . . 4  |-  .x.  =  ( .s `  T )
29 eqid 2283 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
309, 29, 10frlmbasf 27228 . . . . . 6  |-  ( ( I  e.  X  /\  ( U `  Y )  e.  B )  -> 
( U `  Y
) : I --> ( Base `  R ) )
317, 15, 30syl2anc 642 . . . . 5  |-  ( ph  ->  ( U `  Y
) : I --> ( Base `  R ) )
321fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  T )
) )
33 feq3 5377 . . . . . 6  |-  ( (
Base `  R )  =  ( Base `  (Scalar `  T ) )  -> 
( ( U `  Y ) : I --> ( Base `  R
)  <->  ( U `  Y ) : I --> ( Base `  (Scalar `  T ) ) ) )
3432, 33syl 15 . . . . 5  |-  ( ph  ->  ( ( U `  Y ) : I --> ( Base `  R
)  <->  ( U `  Y ) : I --> ( Base `  (Scalar `  T ) ) ) )
3531, 34mpbid 201 . . . 4  |-  ( ph  ->  ( U `  Y
) : I --> ( Base `  (Scalar `  T )
) )
36 frlmup.a . . . 4  |-  ( ph  ->  A : I --> C )
373, 27, 28, 22, 2, 35, 36, 7lcomf 26767 . . 3  |-  ( ph  ->  ( ( U `  Y )  o F 
.x.  A ) : I --> C )
38 ffn 5389 . . . . . . . 8  |-  ( ( U `  Y ) : I --> ( Base `  R )  ->  ( U `  Y )  Fn  I )
3931, 38syl 15 . . . . . . 7  |-  ( ph  ->  ( U `  Y
)  Fn  I )
4039adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( U `  Y
)  Fn  I )
41 ffn 5389 . . . . . . . 8  |-  ( A : I --> C  ->  A  Fn  I )
4236, 41syl 15 . . . . . . 7  |-  ( ph  ->  A  Fn  I )
4342adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  A  Fn  I )
447adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  I  e.  X )
45 eldifi 3298 . . . . . . 7  |-  ( x  e.  ( I  \  { Y } )  ->  x  e.  I )
4645adantl 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  x  e.  I )
47 fnfvof 6090 . . . . . 6  |-  ( ( ( ( U `  Y )  Fn  I  /\  A  Fn  I
)  /\  ( I  e.  X  /\  x  e.  I ) )  -> 
( ( ( U `
 Y )  o F  .x.  A ) `
 x )  =  ( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
) )
4840, 43, 44, 46, 47syl22anc 1183 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y )  o F  .x.  A ) `
 x )  =  ( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
) )
496adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  R  e.  Ring )
5013adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  Y  e.  I )
51 eldifsni 3750 . . . . . . . . . 10  |-  ( x  e.  ( I  \  { Y } )  ->  x  =/=  Y )
5251necomd 2529 . . . . . . . . 9  |-  ( x  e.  ( I  \  { Y } )  ->  Y  =/=  x )
5352adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  Y  =/=  x )
54 eqid 2283 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
558, 49, 44, 50, 46, 53, 54uvcvv0 27239 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  R ) )
561fveq2d 5529 . . . . . . . 8  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5756adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5855, 57eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  (Scalar `  T )
) )
5958oveq1d 5873 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
)  =  ( ( 0g `  (Scalar `  T ) )  .x.  ( A `  x ) ) )
602adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  T  e.  LMod )
61 ffvelrn 5663 . . . . . . 7  |-  ( ( A : I --> C  /\  x  e.  I )  ->  ( A `  x
)  e.  C )
6236, 45, 61syl2an 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( A `  x
)  e.  C )
63 eqid 2283 . . . . . . 7  |-  ( 0g
`  (Scalar `  T )
)  =  ( 0g
`  (Scalar `  T )
)
6422, 3, 28, 63, 23lmod0vs 15663 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( A `  x )  e.  C )  ->  (
( 0g `  (Scalar `  T ) )  .x.  ( A `  x ) )  =  ( 0g
`  T ) )
6560, 62, 64syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( 0g `  (Scalar `  T ) ) 
.x.  ( A `  x ) )  =  ( 0g `  T
) )
6648, 59, 653eqtrd 2319 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y )  o F  .x.  A ) `
 x )  =  ( 0g `  T
) )
6737, 66suppss 5658 . . 3  |-  ( ph  ->  ( `' ( ( U `  Y )  o F  .x.  A
) " ( _V 
\  { ( 0g
`  T ) } ) )  C_  { Y } )
6822, 23, 26, 7, 13, 37, 67gsumpt 15222 . 2  |-  ( ph  ->  ( T  gsumg  ( ( U `  Y )  o F 
.x.  A ) )  =  ( ( ( U `  Y )  o F  .x.  A
) `  Y )
)
69 fnfvof 6090 . . . 4  |-  ( ( ( ( U `  Y )  Fn  I  /\  A  Fn  I
)  /\  ( I  e.  X  /\  Y  e.  I ) )  -> 
( ( ( U `
 Y )  o F  .x.  A ) `
 Y )  =  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
) )
7039, 42, 7, 13, 69syl22anc 1183 . . 3  |-  ( ph  ->  ( ( ( U `
 Y )  o F  .x.  A ) `
 Y )  =  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
) )
71 eqid 2283 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
728, 6, 7, 13, 71uvcvv1 27238 . . . . 5  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  R ) )
731fveq2d 5529 . . . . 5  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  T )
) )
7472, 73eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  (Scalar `  T )
) )
7574oveq1d 5873 . . 3  |-  ( ph  ->  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
)  =  ( ( 1r `  (Scalar `  T ) )  .x.  ( A `  Y ) ) )
76 ffvelrn 5663 . . . . 5  |-  ( ( A : I --> C  /\  Y  e.  I )  ->  ( A `  Y
)  e.  C )
7736, 13, 76syl2anc 642 . . . 4  |-  ( ph  ->  ( A `  Y
)  e.  C )
78 eqid 2283 . . . . 5  |-  ( 1r
`  (Scalar `  T )
)  =  ( 1r
`  (Scalar `  T )
)
7922, 3, 28, 78lmodvs1 15658 . . . 4  |-  ( ( T  e.  LMod  /\  ( A `  Y )  e.  C )  ->  (
( 1r `  (Scalar `  T ) )  .x.  ( A `  Y ) )  =  ( A `
 Y ) )
802, 77, 79syl2anc 642 . . 3  |-  ( ph  ->  ( ( 1r `  (Scalar `  T ) ) 
.x.  ( A `  Y ) )  =  ( A `  Y
) )
8170, 75, 803eqtrd 2319 . 2  |-  ( ph  ->  ( ( ( U `
 Y )  o F  .x.  A ) `
 Y )  =  ( A `  Y
) )
8221, 68, 813eqtrd 2319 1  |-  ( ph  ->  ( E `  ( U `  Y )
)  =  ( A `
 Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640    e. cmpt 4077    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361  CMndccmn 15089   Ringcrg 15337   1rcur 15339   LModclmod 15627   freeLMod cfrlm 27212   unitVec cuvc 27213
This theorem is referenced by:  frlmup3  27252  frlmup4  27253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-sra 15925  df-rgmod 15926  df-dsmm 27198  df-frlm 27214  df-uvc 27215
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