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Theorem frlmup2 18226
Description: The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
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  oF  .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 2442 . . . . . . . 8  |-  (Scalar `  T )  =  (Scalar `  T )
43lmodrng 16955 . . . . . . 7  |-  ( T  e.  LMod  ->  (Scalar `  T )  e.  Ring )
52, 4syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  T )  e.  Ring )
61, 5eqeltrd 2516 . . . . 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 18215 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  U : I --> B )
126, 7, 11syl2anc 661 . . . 4  |-  ( ph  ->  U : I --> B )
13 frlmup.y . . . 4  |-  ( ph  ->  Y  e.  I )
1412, 13ffvelrnd 5843 . . 3  |-  ( ph  ->  ( U `  Y
)  e.  B )
15 oveq1 6097 . . . . 5  |-  ( x  =  ( U `  Y )  ->  (
x  oF  .x.  A )  =  ( ( U `  Y
)  oF  .x.  A ) )
1615oveq2d 6106 . . . 4  |-  ( x  =  ( U `  Y )  ->  ( T  gsumg  ( x  oF  .x.  A ) )  =  ( T  gsumg  ( ( U `  Y )  oF  .x.  A
) ) )
17 frlmup.e . . . 4  |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )
18 ovex 6115 . . . 4  |-  ( T 
gsumg  ( ( U `  Y )  oF  .x.  A ) )  e.  _V
1916, 17, 18fvmpt 5773 . . 3  |-  ( ( U `  Y )  e.  B  ->  ( E `  ( U `  Y ) )  =  ( T  gsumg  ( ( U `  Y )  oF  .x.  A ) ) )
2014, 19syl 16 . 2  |-  ( ph  ->  ( E `  ( U `  Y )
)  =  ( T 
gsumg  ( ( U `  Y )  oF  .x.  A ) ) )
21 frlmup.c . . 3  |-  C  =  ( Base `  T
)
22 eqid 2442 . . 3  |-  ( 0g
`  T )  =  ( 0g `  T
)
23 lmodcmn 16992 . . . 4  |-  ( T  e.  LMod  ->  T  e. CMnd
)
24 cmnmnd 16291 . . . 4  |-  ( T  e. CMnd  ->  T  e.  Mnd )
252, 23, 243syl 20 . . 3  |-  ( ph  ->  T  e.  Mnd )
26 eqid 2442 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
27 frlmup.v . . . 4  |-  .x.  =  ( .s `  T )
28 eqid 2442 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
299, 28, 10frlmbasf 18187 . . . . . 6  |-  ( ( I  e.  X  /\  ( U `  Y )  e.  B )  -> 
( U `  Y
) : I --> ( Base `  R ) )
307, 14, 29syl2anc 661 . . . . 5  |-  ( ph  ->  ( U `  Y
) : I --> ( Base `  R ) )
311fveq2d 5694 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  T )
) )
32 feq3 5543 . . . . . 6  |-  ( (
Base `  R )  =  ( Base `  (Scalar `  T ) )  -> 
( ( U `  Y ) : I --> ( Base `  R
)  <->  ( U `  Y ) : I --> ( Base `  (Scalar `  T ) ) ) )
3331, 32syl 16 . . . . 5  |-  ( ph  ->  ( ( U `  Y ) : I --> ( Base `  R
)  <->  ( U `  Y ) : I --> ( Base `  (Scalar `  T ) ) ) )
3430, 33mpbid 210 . . . 4  |-  ( ph  ->  ( U `  Y
) : I --> ( Base `  (Scalar `  T )
) )
35 frlmup.a . . . 4  |-  ( ph  ->  A : I --> C )
363, 26, 27, 21, 2, 34, 35, 7lcomf 16982 . . 3  |-  ( ph  ->  ( ( U `  Y )  oF  .x.  A ) : I --> C )
37 ffn 5558 . . . . . . . 8  |-  ( ( U `  Y ) : I --> ( Base `  R )  ->  ( U `  Y )  Fn  I )
3830, 37syl 16 . . . . . . 7  |-  ( ph  ->  ( U `  Y
)  Fn  I )
3938adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( U `  Y
)  Fn  I )
40 ffn 5558 . . . . . . . 8  |-  ( A : I --> C  ->  A  Fn  I )
4135, 40syl 16 . . . . . . 7  |-  ( ph  ->  A  Fn  I )
4241adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  A  Fn  I )
437adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  I  e.  X )
44 eldifi 3477 . . . . . . 7  |-  ( x  e.  ( I  \  { Y } )  ->  x  e.  I )
4544adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  x  e.  I )
46 fnfvof 6332 . . . . . 6  |-  ( ( ( ( U `  Y )  Fn  I  /\  A  Fn  I
)  /\  ( I  e.  X  /\  x  e.  I ) )  -> 
( ( ( U `
 Y )  oF  .x.  A ) `
 x )  =  ( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
) )
4739, 42, 43, 45, 46syl22anc 1219 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y )  oF  .x.  A ) `
 x )  =  ( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
) )
486adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  R  e.  Ring )
4913adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  Y  e.  I )
50 eldifsni 4000 . . . . . . . . . 10  |-  ( x  e.  ( I  \  { Y } )  ->  x  =/=  Y )
5150necomd 2694 . . . . . . . . 9  |-  ( x  e.  ( I  \  { Y } )  ->  Y  =/=  x )
5251adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  Y  =/=  x )
53 eqid 2442 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
548, 48, 43, 49, 45, 52, 53uvcvv0 18214 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  R ) )
551fveq2d 5694 . . . . . . . 8  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5655adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5754, 56eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  (Scalar `  T )
) )
5857oveq1d 6105 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
)  =  ( ( 0g `  (Scalar `  T ) )  .x.  ( A `  x ) ) )
592adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  T  e.  LMod )
60 ffvelrn 5840 . . . . . . 7  |-  ( ( A : I --> C  /\  x  e.  I )  ->  ( A `  x
)  e.  C )
6135, 44, 60syl2an 477 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( A `  x
)  e.  C )
62 eqid 2442 . . . . . . 7  |-  ( 0g
`  (Scalar `  T )
)  =  ( 0g
`  (Scalar `  T )
)
6321, 3, 27, 62, 22lmod0vs 16980 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( A `  x )  e.  C )  ->  (
( 0g `  (Scalar `  T ) )  .x.  ( A `  x ) )  =  ( 0g
`  T ) )
6459, 61, 63syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( 0g `  (Scalar `  T ) ) 
.x.  ( A `  x ) )  =  ( 0g `  T
) )
6547, 58, 643eqtrd 2478 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y )  oF  .x.  A ) `
 x )  =  ( 0g `  T
) )
6636, 65suppss 6718 . . 3  |-  ( ph  ->  ( ( ( U `
 Y )  oF  .x.  A ) supp  ( 0g `  T
) )  C_  { Y } )
6721, 22, 25, 7, 13, 36, 66gsumpt 16453 . 2  |-  ( ph  ->  ( T  gsumg  ( ( U `  Y )  oF  .x.  A ) )  =  ( ( ( U `  Y )  oF  .x.  A
) `  Y )
)
68 fnfvof 6332 . . . 4  |-  ( ( ( ( U `  Y )  Fn  I  /\  A  Fn  I
)  /\  ( I  e.  X  /\  Y  e.  I ) )  -> 
( ( ( U `
 Y )  oF  .x.  A ) `
 Y )  =  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
) )
6938, 41, 7, 13, 68syl22anc 1219 . . 3  |-  ( ph  ->  ( ( ( U `
 Y )  oF  .x.  A ) `
 Y )  =  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
) )
70 eqid 2442 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
718, 6, 7, 13, 70uvcvv1 18213 . . . . 5  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  R ) )
721fveq2d 5694 . . . . 5  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  T )
) )
7371, 72eqtrd 2474 . . . 4  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  (Scalar `  T )
) )
7473oveq1d 6105 . . 3  |-  ( ph  ->  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
)  =  ( ( 1r `  (Scalar `  T ) )  .x.  ( A `  Y ) ) )
7535, 13ffvelrnd 5843 . . . 4  |-  ( ph  ->  ( A `  Y
)  e.  C )
76 eqid 2442 . . . . 5  |-  ( 1r
`  (Scalar `  T )
)  =  ( 1r
`  (Scalar `  T )
)
7721, 3, 27, 76lmodvs1 16975 . . . 4  |-  ( ( T  e.  LMod  /\  ( A `  Y )  e.  C )  ->  (
( 1r `  (Scalar `  T ) )  .x.  ( A `  Y ) )  =  ( A `
 Y ) )
782, 75, 77syl2anc 661 . . 3  |-  ( ph  ->  ( ( 1r `  (Scalar `  T ) ) 
.x.  ( A `  Y ) )  =  ( A `  Y
) )
7969, 74, 783eqtrd 2478 . 2  |-  ( ph  ->  ( ( ( U `
 Y )  oF  .x.  A ) `
 Y )  =  ( A `  Y
) )
8020, 67, 793eqtrd 2478 1  |-  ( ph  ->  ( E `  ( U `  Y )
)  =  ( A `
 Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605    \ cdif 3324   {csn 3876    e. cmpt 4349    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090    oFcof 6317   Basecbs 14173  Scalarcsca 14240   .scvsca 14241   0gc0g 14377    gsumg cgsu 14378   Mndcmnd 15408  CMndccmn 16276   1rcur 16602   Ringcrg 16644   LModclmod 16947   freeLMod cfrlm 18170   unitVec cuvc 18206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-fz 11437  df-fzo 11548  df-seq 11806  df-hash 12103  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-hom 14261  df-cco 14262  df-0g 14379  df-gsum 14380  df-prds 14385  df-pws 14387  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-lmod 16949  df-sra 17252  df-rgmod 17253  df-dsmm 18156  df-frlm 18171  df-uvc 18207
This theorem is referenced by:  frlmup3  18227  frlmup4  18228
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