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Theorem lspextmo 17115
Description: A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.)
Hypotheses
Ref Expression
lspextmo.b  |-  B  =  ( Base `  S
)
lspextmo.k  |-  K  =  ( LSpan `  S )
Assertion
Ref Expression
lspextmo  |-  ( ( X  C_  B  /\  ( K `  X )  =  B )  ->  E* g  e.  ( S LMHom  T ) ( g  |`  X )  =  F )
Distinct variable groups:    B, g    g, F    g, K    S, g    T, g    g, X

Proof of Theorem lspextmo
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2460 . . . 4  |-  ( ( ( g  |`  X )  =  F  /\  (
h  |`  X )  =  F )  ->  (
g  |`  X )  =  ( h  |`  X ) )
2 inss1 3567 . . . . . . . . 9  |-  ( g  i^i  h )  C_  g
3 dmss 5035 . . . . . . . . 9  |-  ( ( g  i^i  h ) 
C_  g  ->  dom  ( g  i^i  h
)  C_  dom  g )
42, 3ax-mp 5 . . . . . . . 8  |-  dom  (
g  i^i  h )  C_ 
dom  g
5 lspextmo.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  S
)
6 eqid 2441 . . . . . . . . . . . . 13  |-  ( Base `  T )  =  (
Base `  T )
75, 6lmhmf 17093 . . . . . . . . . . . 12  |-  ( g  e.  ( S LMHom  T
)  ->  g : B
--> ( Base `  T
) )
87ad2antrl 722 . . . . . . . . . . 11  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  g : B
--> ( Base `  T
) )
9 ffn 5556 . . . . . . . . . . 11  |-  ( g : B --> ( Base `  T )  ->  g  Fn  B )
108, 9syl 16 . . . . . . . . . 10  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  g  Fn  B )
1110adantrr 711 . . . . . . . . 9  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  -> 
g  Fn  B )
12 fndm 5507 . . . . . . . . 9  |-  ( g  Fn  B  ->  dom  g  =  B )
1311, 12syl 16 . . . . . . . 8  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  ->  dom  g  =  B
)
144, 13syl5sseq 3401 . . . . . . 7  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  ->  dom  ( g  i^i  h
)  C_  B )
15 simplr 749 . . . . . . . 8  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  -> 
( K `  X
)  =  B )
16 lmhmlmod1 17092 . . . . . . . . . . 11  |-  ( g  e.  ( S LMHom  T
)  ->  S  e.  LMod )
1716adantr 462 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
1817ad2antrl 722 . . . . . . . . 9  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  ->  S  e.  LMod )
19 eqid 2441 . . . . . . . . . . 11  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019lmhmeql 17114 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  dom  ( g  i^i  h )  e.  ( LSubSp `  S )
)
2120ad2antrl 722 . . . . . . . . 9  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  ->  dom  ( g  i^i  h
)  e.  ( LSubSp `  S ) )
22 simprr 751 . . . . . . . . 9  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  ->  X  C_  dom  ( g  i^i  h ) )
23 lspextmo.k . . . . . . . . . 10  |-  K  =  ( LSpan `  S )
2419, 23lspssp 17047 . . . . . . . . 9  |-  ( ( S  e.  LMod  /\  dom  ( g  i^i  h
)  e.  ( LSubSp `  S )  /\  X  C_ 
dom  ( g  i^i  h ) )  -> 
( K `  X
)  C_  dom  ( g  i^i  h ) )
2518, 21, 22, 24syl3anc 1213 . . . . . . . 8  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  -> 
( K `  X
)  C_  dom  ( g  i^i  h ) )
2615, 25eqsstr3d 3388 . . . . . . 7  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  ->  B  C_  dom  ( g  i^i  h ) )
2714, 26eqssd 3370 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( ( g  e.  ( S LMHom  T
)  /\  h  e.  ( S LMHom  T ) )  /\  X  C_  dom  ( g  i^i  h
) ) )  ->  dom  ( g  i^i  h
)  =  B )
2827expr 612 . . . . 5  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  ( X  C_ 
dom  ( g  i^i  h )  ->  dom  ( g  i^i  h
)  =  B ) )
29 simprr 751 . . . . . . 7  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  e.  ( S LMHom  T ) )
305, 6lmhmf 17093 . . . . . . 7  |-  ( h  e.  ( S LMHom  T
)  ->  h : B
--> ( Base `  T
) )
31 ffn 5556 . . . . . . 7  |-  ( h : B --> ( Base `  T )  ->  h  Fn  B )
3229, 30, 313syl 20 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  Fn  B )
33 simpll 748 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  X  C_  B
)
34 fnreseql 5810 . . . . . 6  |-  ( ( g  Fn  B  /\  h  Fn  B  /\  X  C_  B )  -> 
( ( g  |`  X )  =  ( h  |`  X )  <->  X 
C_  dom  ( g  i^i  h ) ) )
3510, 32, 33, 34syl3anc 1213 . . . . 5  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  ( (
g  |`  X )  =  ( h  |`  X )  <-> 
X  C_  dom  ( g  i^i  h ) ) )
36 fneqeql 5808 . . . . . 6  |-  ( ( g  Fn  B  /\  h  Fn  B )  ->  ( g  =  h  <->  dom  ( g  i^i  h
)  =  B ) )
3710, 32, 36syl2anc 656 . . . . 5  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  ( g  =  h  <->  dom  ( g  i^i  h )  =  B ) )
3828, 35, 373imtr4d 268 . . . 4  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  ( (
g  |`  X )  =  ( h  |`  X )  ->  g  =  h ) )
391, 38syl5 32 . . 3  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  ( (
( g  |`  X )  =  F  /\  (
h  |`  X )  =  F )  ->  g  =  h ) )
4039ralrimivva 2806 . 2  |-  ( ( X  C_  B  /\  ( K `  X )  =  B )  ->  A. g  e.  ( S LMHom  T ) A. h  e.  ( S LMHom  T ) ( ( ( g  |`  X )  =  F  /\  ( h  |`  X )  =  F )  ->  g  =  h ) )
41 reseq1 5100 . . . 4  |-  ( g  =  h  ->  (
g  |`  X )  =  ( h  |`  X ) )
4241eqeq1d 2449 . . 3  |-  ( g  =  h  ->  (
( g  |`  X )  =  F  <->  ( h  |`  X )  =  F ) )
4342rmo4 3149 . 2  |-  ( E* g  e.  ( S LMHom 
T ) ( g  |`  X )  =  F  <->  A. g  e.  ( S LMHom  T ) A. h  e.  ( S LMHom  T ) ( ( ( g  |`  X )  =  F  /\  ( h  |`  X )  =  F )  ->  g  =  h ) )
4440, 43sylibr 212 1  |-  ( ( X  C_  B  /\  ( K `  X )  =  B )  ->  E* g  e.  ( S LMHom  T ) ( g  |`  X )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E*wrmo 2716    i^i cin 3324    C_ wss 3325   dom cdm 4836    |` cres 4838    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   LModclmod 16928   LSubSpclss 16991   LSpanclspn 17030   LMHom clmhm 17078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-ghm 15738  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-lss 16992  df-lsp 17031  df-lmhm 17081
This theorem is referenced by:  frlmup4  18188
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