MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frlmup2 Structured version   Unicode version

Theorem frlmup2 18706
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 2443 . . . . . . . 8  |-  (Scalar `  T )  =  (Scalar `  T )
43lmodring 17394 . . . . . . 7  |-  ( T  e.  LMod  ->  (Scalar `  T )  e.  Ring )
52, 4syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  T )  e.  Ring )
61, 5eqeltrd 2531 . . . . 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 18695 . . . . 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 6017 . . 3  |-  ( ph  ->  ( U `  Y
)  e.  B )
15 oveq1 6288 . . . . 5  |-  ( x  =  ( U `  Y )  ->  (
x  oF  .x.  A )  =  ( ( U `  Y
)  oF  .x.  A ) )
1615oveq2d 6297 . . . 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 6309 . . . 4  |-  ( T 
gsumg  ( ( U `  Y )  oF  .x.  A ) )  e.  _V
1916, 17, 18fvmpt 5941 . . 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 2443 . . 3  |-  ( 0g
`  T )  =  ( 0g `  T
)
23 lmodcmn 17432 . . . 4  |-  ( T  e.  LMod  ->  T  e. CMnd
)
24 cmnmnd 16687 . . . 4  |-  ( T  e. CMnd  ->  T  e.  Mnd )
252, 23, 243syl 20 . . 3  |-  ( ph  ->  T  e.  Mnd )
26 eqid 2443 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
27 frlmup.v . . . 4  |-  .x.  =  ( .s `  T )
28 eqid 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
299, 28, 10frlmbasf 18667 . . . . . 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 5860 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  T )
) )
3231feq3d 5709 . . . . 5  |-  ( ph  ->  ( ( U `  Y ) : I --> ( Base `  R
)  <->  ( U `  Y ) : I --> ( Base `  (Scalar `  T ) ) ) )
3330, 32mpbid 210 . . . 4  |-  ( ph  ->  ( U `  Y
) : I --> ( Base `  (Scalar `  T )
) )
34 frlmup.a . . . 4  |-  ( ph  ->  A : I --> C )
353, 26, 27, 21, 2, 33, 34, 7lcomf 17422 . . 3  |-  ( ph  ->  ( ( U `  Y )  oF  .x.  A ) : I --> C )
36 ffn 5721 . . . . . . . 8  |-  ( ( U `  Y ) : I --> ( Base `  R )  ->  ( U `  Y )  Fn  I )
3730, 36syl 16 . . . . . . 7  |-  ( ph  ->  ( U `  Y
)  Fn  I )
3837adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( U `  Y
)  Fn  I )
39 ffn 5721 . . . . . . . 8  |-  ( A : I --> C  ->  A  Fn  I )
4034, 39syl 16 . . . . . . 7  |-  ( ph  ->  A  Fn  I )
4140adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  A  Fn  I )
427adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  I  e.  X )
43 eldifi 3611 . . . . . . 7  |-  ( x  e.  ( I  \  { Y } )  ->  x  e.  I )
4443adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  x  e.  I )
45 fnfvof 6538 . . . . . 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 )
) )
4638, 41, 42, 44, 45syl22anc 1230 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y )  oF  .x.  A ) `
 x )  =  ( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
) )
476adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  R  e.  Ring )
4813adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  Y  e.  I )
49 eldifsni 4141 . . . . . . . . . 10  |-  ( x  e.  ( I  \  { Y } )  ->  x  =/=  Y )
5049necomd 2714 . . . . . . . . 9  |-  ( x  e.  ( I  \  { Y } )  ->  Y  =/=  x )
5150adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  Y  =/=  x )
52 eqid 2443 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
538, 47, 42, 48, 44, 51, 52uvcvv0 18694 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  R ) )
541fveq2d 5860 . . . . . . . 8  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5554adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
5653, 55eqtrd 2484 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( U `  Y ) `  x
)  =  ( 0g
`  (Scalar `  T )
) )
5756oveq1d 6296 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y ) `  x )  .x.  ( A `  x )
)  =  ( ( 0g `  (Scalar `  T ) )  .x.  ( A `  x ) ) )
582adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  ->  T  e.  LMod )
59 ffvelrn 6014 . . . . . . 7  |-  ( ( A : I --> C  /\  x  e.  I )  ->  ( A `  x
)  e.  C )
6034, 43, 59syl2an 477 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( A `  x
)  e.  C )
61 eqid 2443 . . . . . . 7  |-  ( 0g
`  (Scalar `  T )
)  =  ( 0g
`  (Scalar `  T )
)
6221, 3, 27, 61, 22lmod0vs 17419 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( A `  x )  e.  C )  ->  (
( 0g `  (Scalar `  T ) )  .x.  ( A `  x ) )  =  ( 0g
`  T ) )
6358, 60, 62syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( 0g `  (Scalar `  T ) ) 
.x.  ( A `  x ) )  =  ( 0g `  T
) )
6446, 57, 633eqtrd 2488 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  { Y } ) )  -> 
( ( ( U `
 Y )  oF  .x.  A ) `
 x )  =  ( 0g `  T
) )
6535, 64suppss 6932 . . 3  |-  ( ph  ->  ( ( ( U `
 Y )  oF  .x.  A ) supp  ( 0g `  T
) )  C_  { Y } )
6621, 22, 25, 7, 13, 35, 65gsumpt 16862 . 2  |-  ( ph  ->  ( T  gsumg  ( ( U `  Y )  oF  .x.  A ) )  =  ( ( ( U `  Y )  oF  .x.  A
) `  Y )
)
67 fnfvof 6538 . . . 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 )
) )
6837, 40, 7, 13, 67syl22anc 1230 . . 3  |-  ( ph  ->  ( ( ( U `
 Y )  oF  .x.  A ) `
 Y )  =  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
) )
69 eqid 2443 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
708, 6, 7, 13, 69uvcvv1 18693 . . . . 5  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  R ) )
711fveq2d 5860 . . . . 5  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  T )
) )
7270, 71eqtrd 2484 . . . 4  |-  ( ph  ->  ( ( U `  Y ) `  Y
)  =  ( 1r
`  (Scalar `  T )
) )
7372oveq1d 6296 . . 3  |-  ( ph  ->  ( ( ( U `
 Y ) `  Y )  .x.  ( A `  Y )
)  =  ( ( 1r `  (Scalar `  T ) )  .x.  ( A `  Y ) ) )
7434, 13ffvelrnd 6017 . . . 4  |-  ( ph  ->  ( A `  Y
)  e.  C )
75 eqid 2443 . . . . 5  |-  ( 1r
`  (Scalar `  T )
)  =  ( 1r
`  (Scalar `  T )
)
7621, 3, 27, 75lmodvs1 17414 . . . 4  |-  ( ( T  e.  LMod  /\  ( A `  Y )  e.  C )  ->  (
( 1r `  (Scalar `  T ) )  .x.  ( A `  Y ) )  =  ( A `
 Y ) )
772, 74, 76syl2anc 661 . . 3  |-  ( ph  ->  ( ( 1r `  (Scalar `  T ) ) 
.x.  ( A `  Y ) )  =  ( A `  Y
) )
7868, 73, 773eqtrd 2488 . 2  |-  ( ph  ->  ( ( ( U `
 Y )  oF  .x.  A ) `
 Y )  =  ( A `  Y
) )
7920, 66, 783eqtrd 2488 1  |-  ( ph  ->  ( E `  ( U `  Y )
)  =  ( A `
 Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638    \ cdif 3458   {csn 4014    |-> cmpt 4495    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   0gc0g 14714    gsumg cgsu 14715   Mndcmnd 15793  CMndccmn 16672   1rcur 17027   Ringcrg 17072   LModclmod 17386   freeLMod cfrlm 18650   unitVec cuvc 18686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-fzo 11804  df-seq 12087  df-hash 12385  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-hom 14598  df-cco 14599  df-0g 14716  df-gsum 14717  df-prds 14722  df-pws 14724  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-grp 15931  df-minusg 15932  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-lmod 17388  df-sra 17692  df-rgmod 17693  df-dsmm 18636  df-frlm 18651  df-uvc 18687
This theorem is referenced by:  frlmup3  18707  frlmup4  18708
  Copyright terms: Public domain W3C validator