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Theorem frlmup2 27354
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 2296 . . . . . . . 8  |-  (Scalar `  T )  =  (Scalar `  T )
43lmodrng 15651 . . . . . . 7  |-  ( T  e.  LMod  ->  (Scalar `  T )  e.  Ring )
52, 4syl 15 . . . . . 6  |-  ( ph  ->  (Scalar `  T )  e.  Ring )
61, 5eqeltrd 2370 . . . . 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 27343 . . . . 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 5679 . . . 4  |-  ( ( U : I --> B  /\  Y  e.  I )  ->  ( U `  Y
)  e.  B )
1512, 13, 14syl2anc 642 . . 3  |-  ( ph  ->  ( U `  Y
)  e.  B )
16 oveq1 5881 . . . . 5  |-  ( x  =  ( U `  Y )  ->  (
x  o F  .x.  A )  =  ( ( U `  Y
)  o F  .x.  A ) )
1716oveq2d 5890 . . . 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 5899 . . . 4  |-  ( T 
gsumg  ( ( U `  Y )  o F 
.x.  A ) )  e.  _V
2017, 18, 19fvmpt 5618 . . 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 2296 . . 3  |-  ( 0g
`  T )  =  ( 0g `  T
)
24 lmodcmn 15689 . . . 4  |-  ( T  e.  LMod  ->  T  e. CMnd
)
25 cmnmnd 15120 . . . 4  |-  ( T  e. CMnd  ->  T  e.  Mnd )
262, 24, 253syl 18 . . 3  |-  ( ph  ->  T  e.  Mnd )
27 eqid 2296 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
28 frlmup.v . . . 4  |-  .x.  =  ( .s `  T )
29 eqid 2296 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
309, 29, 10frlmbasf 27331 . . . . . 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 5545 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  T )
) )
33 feq3 5393 . . . . . 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 26870 . . 3  |-  ( ph  ->  ( ( U `  Y )  o F 
.x.  A ) : I --> C )
38 ffn 5405 . . . . . . . 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 5405 . . . . . . . 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 3311 . . . . . . 7  |-  ( x  e.  ( I  \  { Y } )  ->  x  e.  I )
4645adantl 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  x  e.  I )
47 fnfvof 6106 . . . . . 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 3763 . . . . . . . . . 10  |-  ( x  e.  ( I  \  { Y } )  ->  x  =/=  Y )
5251necomd 2542 . . . . . . . . 9  |-  ( x  e.  ( I  \  { Y } )  ->  Y  =/=  x )
5352adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  Y  =/=  x )
54 eqid 2296 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
558, 49, 44, 50, 46, 53, 54uvcvv0 27342 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  R ) )
561fveq2d 5545 . . . . . . . 8  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5756adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5855, 57eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  (Scalar `  T )
) )
5958oveq1d 5889 . . . . 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 5679 . . . . . . 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 2296 . . . . . . 7  |-  ( 0g
`  (Scalar `  T )
)  =  ( 0g
`  (Scalar `  T )
)
6422, 3, 28, 63, 23lmod0vs 15679 . . . . . 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 2332 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y )  o F  .x.  A ) `
 x )  =  ( 0g `  T
) )
6737, 66suppss 5674 . . 3  |-  ( ph  ->  ( `' ( ( U `  Y )  o F  .x.  A
) " ( _V 
\  { ( 0g
`  T ) } ) )  C_  { Y } )
6822, 23, 26, 7, 13, 37, 67gsumpt 15238 . 2  |-  ( ph  ->  ( T  gsumg  ( ( U `  Y )  o F 
.x.  A ) )  =  ( ( ( U `  Y )  o F  .x.  A
) `  Y )
)
69 fnfvof 6106 . . . 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 2296 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
728, 6, 7, 13, 71uvcvv1 27341 . . . . 5  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  R ) )
731fveq2d 5545 . . . . 5  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  T )
) )
7472, 73eqtrd 2328 . . . 4  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  (Scalar `  T )
) )
7574oveq1d 5889 . . 3  |-  ( ph  ->  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
)  =  ( ( 1r `  (Scalar `  T ) )  .x.  ( A `  Y ) ) )
76 ffvelrn 5679 . . . . 5  |-  ( ( A : I --> C  /\  Y  e.  I )  ->  ( A `  Y
)  e.  C )
7736, 13, 76syl2anc 642 . . . 4  |-  ( ph  ->  ( A `  Y
)  e.  C )
78 eqid 2296 . . . . 5  |-  ( 1r
`  (Scalar `  T )
)  =  ( 1r
`  (Scalar `  T )
)
7922, 3, 28, 78lmodvs1 15674 . . . 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 2332 . 2  |-  ( ph  ->  ( ( ( U `
 Y )  o F  .x.  A ) `
 Y )  =  ( A `  Y
) )
8221, 68, 813eqtrd 2332 1  |-  ( ph  ->  ( E `  ( U `  Y )
)  =  ( A `
 Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653    e. cmpt 4093    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377  CMndccmn 15105   Ringcrg 15353   1rcur 15355   LModclmod 15643   freeLMod cfrlm 27315   unitVec cuvc 27316
This theorem is referenced by:  frlmup3  27355  frlmup4  27356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-sra 15941  df-rgmod 15942  df-dsmm 27301  df-frlm 27317  df-uvc 27318
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