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Theorem frlmup3 26575
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 26573 . . 3  |-  ( ph  ->  E  e.  ( F LMHom 
T ) )
11 eqid 2358 . . . . . . . 8  |-  (Scalar `  T )  =  (Scalar `  T )
1211lmodrng 15728 . . . . . . 7  |-  ( T  e.  LMod  ->  (Scalar `  T )  e.  Ring )
136, 12syl 15 . . . . . 6  |-  ( ph  ->  (Scalar `  T )  e.  Ring )
148, 13eqeltrd 2432 . . . . 5  |-  ( ph  ->  R  e.  Ring )
15 eqid 2358 . . . . . 6  |-  ( R unitVec  I )  =  ( R unitVec  I )
1615, 1, 2uvcff 26563 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  ( R unitVec  I ) : I --> B )
1714, 7, 16syl2anc 642 . . . 4  |-  ( ph  ->  ( R unitVec  I ) : I --> B )
18 frn 5475 . . . 4  |-  ( ( R unitVec  I ) : I --> B  ->  ran  ( R unitVec  I )  C_  B )
1917, 18syl 15 . . 3  |-  ( ph  ->  ran  ( R unitVec  I
)  C_  B )
20 eqid 2358 . . . 4  |-  ( LSpan `  F )  =  (
LSpan `  F )
21 frlmup.k . . . 4  |-  K  =  ( LSpan `  T )
222, 20, 21lmhmlsp 15899 . . 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 642 . 2  |-  ( ph  ->  ( E " (
( LSpan `  F ) `  ran  ( R unitVec  I
) ) )  =  ( K `  ( E " ran  ( R unitVec  I ) ) ) )
242, 3lmhmf 15884 . . . . . 6  |-  ( E  e.  ( F LMHom  T
)  ->  E : B
--> C )
2510, 24syl 15 . . . . 5  |-  ( ph  ->  E : B --> C )
26 ffn 5469 . . . . 5  |-  ( E : B --> C  ->  E  Fn  B )
2725, 26syl 15 . . . 4  |-  ( ph  ->  E  Fn  B )
28 fnima 5441 . . . 4  |-  ( E  Fn  B  ->  ( E " B )  =  ran  E )
2927, 28syl 15 . . 3  |-  ( ph  ->  ( E " B
)  =  ran  E
)
30 eqid 2358 . . . . . . . 8  |-  (LBasis `  F )  =  (LBasis `  F )
311, 15, 30frlmlbs 26572 . . . . . . 7  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  ran  ( R unitVec  I )  e.  (LBasis `  F )
)
3214, 7, 31syl2anc 642 . . . . . 6  |-  ( ph  ->  ran  ( R unitVec  I
)  e.  (LBasis `  F ) )
332, 30, 20lbssp 15925 . . . . . 6  |-  ( ran  ( R unitVec  I )  e.  (LBasis `  F )  ->  ( ( LSpan `  F
) `  ran  ( R unitVec  I ) )  =  B )
3432, 33syl 15 . . . . 5  |-  ( ph  ->  ( ( LSpan `  F
) `  ran  ( R unitVec  I ) )  =  B )
3534eqcomd 2363 . . . 4  |-  ( ph  ->  B  =  ( (
LSpan `  F ) `  ran  ( R unitVec  I )
) )
3635imaeq2d 5091 . . 3  |-  ( ph  ->  ( E " B
)  =  ( E
" ( ( LSpan `  F ) `  ran  ( R unitVec  I ) ) ) )
3729, 36eqtr3d 2392 . 2  |-  ( ph  ->  ran  E  =  ( E " ( (
LSpan `  F ) `  ran  ( R unitVec  I )
) ) )
38 imaco 5257 . . . 4  |-  ( ( E  o.  ( R unitVec  I ) ) "
I )  =  ( E " ( ( R unitVec  I ) "
I ) )
39 ffn 5469 . . . . . . . 8  |-  ( A : I --> C  ->  A  Fn  I )
409, 39syl 15 . . . . . . 7  |-  ( ph  ->  A  Fn  I )
41 ffn 5469 . . . . . . . . 9  |-  ( ( R unitVec  I ) : I --> B  ->  ( R unitVec  I )  Fn  I
)
4217, 41syl 15 . . . . . . . 8  |-  ( ph  ->  ( R unitVec  I )  Fn  I )
43 fnco 5431 . . . . . . . 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 1182 . . . . . . 7  |-  ( ph  ->  ( E  o.  ( R unitVec  I ) )  Fn  I )
45 fvco2 5674 . . . . . . . . 9  |-  ( ( ( R unitVec  I )  Fn  I  /\  u  e.  I )  ->  (
( E  o.  ( R unitVec  I ) ) `  u )  =  ( E `  ( ( R unitVec  I ) `  u ) ) )
4642, 45sylan 457 . . . . . . . 8  |-  ( (
ph  /\  u  e.  I )  ->  (
( E  o.  ( R unitVec  I ) ) `  u )  =  ( E `  ( ( R unitVec  I ) `  u ) ) )
476adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  T  e.  LMod )
487adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  I  e.  X )
498adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  R  =  (Scalar `  T )
)
509adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  A : I --> C )
51 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  I )  ->  u  e.  I )
521, 2, 3, 4, 5, 47, 48, 49, 50, 51, 15frlmup2 26574 . . . . . . . 8  |-  ( (
ph  /\  u  e.  I )  ->  ( E `  ( ( R unitVec  I ) `  u
) )  =  ( A `  u ) )
5346, 52eqtr2d 2391 . . . . . . 7  |-  ( (
ph  /\  u  e.  I )  ->  ( A `  u )  =  ( ( E  o.  ( R unitVec  I
) ) `  u
) )
5440, 44, 53eqfnfvd 5705 . . . . . 6  |-  ( ph  ->  A  =  ( E  o.  ( R unitVec  I
) ) )
5554imaeq1d 5090 . . . . 5  |-  ( ph  ->  ( A " I
)  =  ( ( E  o.  ( R unitVec  I ) ) "
I ) )
56 fnima 5441 . . . . . 6  |-  ( A  Fn  I  ->  ( A " I )  =  ran  A )
5740, 56syl 15 . . . . 5  |-  ( ph  ->  ( A " I
)  =  ran  A
)
5855, 57eqtr3d 2392 . . . 4  |-  ( ph  ->  ( ( E  o.  ( R unitVec  I ) )
" I )  =  ran  A )
59 fnima 5441 . . . . . 6  |-  ( ( R unitVec  I )  Fn  I  ->  ( ( R unitVec  I ) " I
)  =  ran  ( R unitVec  I ) )
6042, 59syl 15 . . . . 5  |-  ( ph  ->  ( ( R unitVec  I
) " I )  =  ran  ( R unitVec  I ) )
6160imaeq2d 5091 . . . 4  |-  ( ph  ->  ( E " (
( R unitVec  I ) " I ) )  =  ( E " ran  ( R unitVec  I )
) )
6238, 58, 613eqtr3a 2414 . . 3  |-  ( ph  ->  ran  A  =  ( E " ran  ( R unitVec  I ) ) )
6362fveq2d 5609 . 2  |-  ( ph  ->  ( K `  ran  A )  =  ( K `
 ( E " ran  ( R unitVec  I )
) ) )
6423, 37, 633eqtr4d 2400 1  |-  ( ph  ->  ran  E  =  ( K `  ran  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228    e. cmpt 4156   ran crn 4769   "cima 4771    o. ccom 4772    Fn wfn 5329   -->wf 5330   ` cfv 5334  (class class class)co 5942    o Fcof 6160   Basecbs 13239  Scalarcsca 13302   .scvsca 13303    gsumg cgsu 13494   Ringcrg 15430   LModclmod 15720   LSpanclspn 15821   LMHom clmhm 15869  LBasisclbs 15920   freeLMod cfrlm 26535   unitVec cuvc 26536
This theorem is referenced by:  ellspd  26577  indlcim  26633  lnrfg  26646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-fz 10872  df-fzo 10960  df-seq 11136  df-hash 11428  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-hom 13323  df-cco 13324  df-prds 13441  df-pws 13443  df-0g 13497  df-gsum 13498  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-mhm 14508  df-submnd 14509  df-grp 14582  df-minusg 14583  df-sbg 14584  df-mulg 14585  df-subg 14711  df-ghm 14774  df-cntz 14886  df-cmn 15184  df-abl 15185  df-mgp 15419  df-rng 15433  df-ur 15435  df-subrg 15636  df-lmod 15722  df-lss 15783  df-lsp 15822  df-lmhm 15872  df-lbs 15921  df-sra 16018  df-rgmod 16019  df-nzr 16103  df-dsmm 26521  df-frlm 26537  df-uvc 26538
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