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Theorem frlmup3 27243
Description: The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-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.k  |-  K  =  ( LSpan `  T )
Assertion
Ref Expression
frlmup3  |-  ( ph  ->  ran  E  =  ( K `  ran  A
) )
Distinct variable groups:    x, R    x, I    x, F    x, B    x, C    x,  .x.    x, A    x, X    x, K    ph, x    x, T
Allowed substitution hint:    E( x)

Proof of Theorem frlmup3
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 frlmup.f . . . 4  |-  F  =  ( R freeLMod  I )
2 frlmup.b . . . 4  |-  B  =  ( Base `  F
)
3 frlmup.c . . . 4  |-  C  =  ( Base `  T
)
4 frlmup.v . . . 4  |-  .x.  =  ( .s `  T )
5 frlmup.e . . . 4  |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F 
.x.  A ) ) )
6 frlmup.t . . . 4  |-  ( ph  ->  T  e.  LMod )
7 frlmup.i . . . 4  |-  ( ph  ->  I  e.  X )
8 frlmup.r . . . 4  |-  ( ph  ->  R  =  (Scalar `  T ) )
9 frlmup.a . . . 4  |-  ( ph  ->  A : I --> C )
101, 2, 3, 4, 5, 6, 7, 8, 9frlmup1 27241 . . 3  |-  ( ph  ->  E  e.  ( F LMHom 
T ) )
11 eqid 2438 . . . . . . . 8  |-  (Scalar `  T )  =  (Scalar `  T )
1211lmodrng 15963 . . . . . . 7  |-  ( T  e.  LMod  ->  (Scalar `  T )  e.  Ring )
136, 12syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  T )  e.  Ring )
148, 13eqeltrd 2512 . . . . 5  |-  ( ph  ->  R  e.  Ring )
15 eqid 2438 . . . . . 6  |-  ( R unitVec  I )  =  ( R unitVec  I )
1615, 1, 2uvcff 27231 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  ( R unitVec  I ) : I --> B )
1714, 7, 16syl2anc 644 . . . 4  |-  ( ph  ->  ( R unitVec  I ) : I --> B )
18 frn 5600 . . . 4  |-  ( ( R unitVec  I ) : I --> B  ->  ran  ( R unitVec  I )  C_  B )
1917, 18syl 16 . . 3  |-  ( ph  ->  ran  ( R unitVec  I
)  C_  B )
20 eqid 2438 . . . 4  |-  ( LSpan `  F )  =  (
LSpan `  F )
21 frlmup.k . . . 4  |-  K  =  ( LSpan `  T )
222, 20, 21lmhmlsp 16130 . . 3  |-  ( ( E  e.  ( F LMHom 
T )  /\  ran  ( R unitVec  I )  C_  B )  ->  ( E " ( ( LSpan `  F ) `  ran  ( R unitVec  I ) ) )  =  ( K `
 ( E " ran  ( R unitVec  I )
) ) )
2310, 19, 22syl2anc 644 . 2  |-  ( ph  ->  ( E " (
( LSpan `  F ) `  ran  ( R unitVec  I
) ) )  =  ( K `  ( E " ran  ( R unitVec  I ) ) ) )
242, 3lmhmf 16115 . . . . . 6  |-  ( E  e.  ( F LMHom  T
)  ->  E : B
--> C )
2510, 24syl 16 . . . . 5  |-  ( ph  ->  E : B --> C )
26 ffn 5594 . . . . 5  |-  ( E : B --> C  ->  E  Fn  B )
2725, 26syl 16 . . . 4  |-  ( ph  ->  E  Fn  B )
28 fnima 5566 . . . 4  |-  ( E  Fn  B  ->  ( E " B )  =  ran  E )
2927, 28syl 16 . . 3  |-  ( ph  ->  ( E " B
)  =  ran  E
)
30 eqid 2438 . . . . . . . 8  |-  (LBasis `  F )  =  (LBasis `  F )
311, 15, 30frlmlbs 27240 . . . . . . 7  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  ran  ( R unitVec  I )  e.  (LBasis `  F )
)
3214, 7, 31syl2anc 644 . . . . . 6  |-  ( ph  ->  ran  ( R unitVec  I
)  e.  (LBasis `  F ) )
332, 30, 20lbssp 16156 . . . . . 6  |-  ( ran  ( R unitVec  I )  e.  (LBasis `  F )  ->  ( ( LSpan `  F
) `  ran  ( R unitVec  I ) )  =  B )
3432, 33syl 16 . . . . 5  |-  ( ph  ->  ( ( LSpan `  F
) `  ran  ( R unitVec  I ) )  =  B )
3534eqcomd 2443 . . . 4  |-  ( ph  ->  B  =  ( (
LSpan `  F ) `  ran  ( R unitVec  I )
) )
3635imaeq2d 5206 . . 3  |-  ( ph  ->  ( E " B
)  =  ( E
" ( ( LSpan `  F ) `  ran  ( R unitVec  I ) ) ) )
3729, 36eqtr3d 2472 . 2  |-  ( ph  ->  ran  E  =  ( E " ( (
LSpan `  F ) `  ran  ( R unitVec  I )
) ) )
38 imaco 5378 . . . 4  |-  ( ( E  o.  ( R unitVec  I ) ) "
I )  =  ( E " ( ( R unitVec  I ) "
I ) )
39 ffn 5594 . . . . . . . 8  |-  ( A : I --> C  ->  A  Fn  I )
409, 39syl 16 . . . . . . 7  |-  ( ph  ->  A  Fn  I )
41 ffn 5594 . . . . . . . . 9  |-  ( ( R unitVec  I ) : I --> B  ->  ( R unitVec  I )  Fn  I
)
4217, 41syl 16 . . . . . . . 8  |-  ( ph  ->  ( R unitVec  I )  Fn  I )
43 fnco 5556 . . . . . . . 8  |-  ( ( E  Fn  B  /\  ( R unitVec  I )  Fn  I  /\  ran  ( R unitVec  I )  C_  B
)  ->  ( E  o.  ( R unitVec  I )
)  Fn  I )
4427, 42, 19, 43syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( E  o.  ( R unitVec  I ) )  Fn  I )
45 fvco2 5801 . . . . . . . . 9  |-  ( ( ( R unitVec  I )  Fn  I  /\  u  e.  I )  ->  (
( E  o.  ( R unitVec  I ) ) `  u )  =  ( E `  ( ( R unitVec  I ) `  u ) ) )
4642, 45sylan 459 . . . . . . . 8  |-  ( (
ph  /\  u  e.  I )  ->  (
( E  o.  ( R unitVec  I ) ) `  u )  =  ( E `  ( ( R unitVec  I ) `  u ) ) )
476adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  T  e.  LMod )
487adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  I  e.  X )
498adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  R  =  (Scalar `  T )
)
509adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  A : I --> C )
51 simpr 449 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  u  e.  I )
521, 2, 3, 4, 5, 47, 48, 49, 50, 51, 15frlmup2 27242 . . . . . . . 8  |-  ( (
ph  /\  u  e.  I )  ->  ( E `  ( ( R unitVec  I ) `  u
) )  =  ( A `  u ) )
5346, 52eqtr2d 2471 . . . . . . 7  |-  ( (
ph  /\  u  e.  I )  ->  ( A `  u )  =  ( ( E  o.  ( R unitVec  I
) ) `  u
) )
5440, 44, 53eqfnfvd 5833 . . . . . 6  |-  ( ph  ->  A  =  ( E  o.  ( R unitVec  I
) ) )
5554imaeq1d 5205 . . . . 5  |-  ( ph  ->  ( A " I
)  =  ( ( E  o.  ( R unitVec  I ) ) "
I ) )
56 fnima 5566 . . . . . 6  |-  ( A  Fn  I  ->  ( A " I )  =  ran  A )
5740, 56syl 16 . . . . 5  |-  ( ph  ->  ( A " I
)  =  ran  A
)
5855, 57eqtr3d 2472 . . . 4  |-  ( ph  ->  ( ( E  o.  ( R unitVec  I ) )
" I )  =  ran  A )
59 fnima 5566 . . . . . 6  |-  ( ( R unitVec  I )  Fn  I  ->  ( ( R unitVec  I ) " I
)  =  ran  ( R unitVec  I ) )
6042, 59syl 16 . . . . 5  |-  ( ph  ->  ( ( R unitVec  I
) " I )  =  ran  ( R unitVec  I ) )
6160imaeq2d 5206 . . . 4  |-  ( ph  ->  ( E " (
( R unitVec  I ) " I ) )  =  ( E " ran  ( R unitVec  I )
) )
6238, 58, 613eqtr3a 2494 . . 3  |-  ( ph  ->  ran  A  =  ( E " ran  ( R unitVec  I ) ) )
6362fveq2d 5735 . 2  |-  ( ph  ->  ( K `  ran  A )  =  ( K `
 ( E " ran  ( R unitVec  I )
) ) )
6423, 37, 633eqtr4d 2480 1  |-  ( ph  ->  ran  E  =  ( K `  ran  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322    e. cmpt 4269   ran crn 4882   "cima 4884    o. ccom 4885    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   Basecbs 13474  Scalarcsca 13537   .scvsca 13538    gsumg cgsu 13729   Ringcrg 15665   LModclmod 15955   LSpanclspn 16052   LMHom clmhm 16100  LBasisclbs 16151   freeLMod cfrlm 27203   unitVec cuvc 27204
This theorem is referenced by:  ellspd  27245  indlcim  27301  lnrfg  27314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-hom 13558  df-cco 13559  df-prds 13676  df-pws 13678  df-0g 13732  df-gsum 13733  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-ghm 15009  df-cntz 15121  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-ur 15670  df-subrg 15871  df-lmod 15957  df-lss 16014  df-lsp 16053  df-lmhm 16103  df-lbs 16152  df-sra 16249  df-rgmod 16250  df-nzr 16334  df-dsmm 27189  df-frlm 27205  df-uvc 27206
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