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Theorem lspextmo 16087
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 2423 . . . 4  |-  ( ( ( g  |`  X )  =  F  /\  (
h  |`  X )  =  F )  ->  (
g  |`  X )  =  ( h  |`  X ) )
2 inss1 3521 . . . . . . . . 9  |-  ( g  i^i  h )  C_  g
3 dmss 5028 . . . . . . . . 9  |-  ( ( g  i^i  h ) 
C_  g  ->  dom  ( g  i^i  h
)  C_  dom  g )
42, 3ax-mp 8 . . . . . . . 8  |-  dom  (
g  i^i  h )  C_ 
dom  g
5 lspextmo.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  S
)
6 eqid 2404 . . . . . . . . . . . . 13  |-  ( Base `  T )  =  (
Base `  T )
75, 6lmhmf 16065 . . . . . . . . . . . 12  |-  ( g  e.  ( S LMHom  T
)  ->  g : B
--> ( Base `  T
) )
87ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  g : B
--> ( Base `  T
) )
9 ffn 5550 . . . . . . . . . . 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 698 . . . . . . . . 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 5503 . . . . . . . . 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 3356 . . . . . . 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 732 . . . . . . . 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 16064 . . . . . . . . . . 11  |-  ( g  e.  ( S LMHom  T
)  ->  S  e.  LMod )
1716adantr 452 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
1817ad2antrl 709 . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019lmhmeql 16086 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  dom  ( g  i^i  h )  e.  ( LSubSp `  S )
)
2120ad2antrl 709 . . . . . . . . 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 734 . . . . . . . . 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 16019 . . . . . . . . 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 1184 . . . . . . . 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 3343 . . . . . . 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 3325 . . . . . 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 599 . . . . 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 734 . . . . . . 7  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  e.  ( S LMHom  T ) )
305, 6lmhmf 16065 . . . . . . 7  |-  ( h  e.  ( S LMHom  T
)  ->  h : B
--> ( Base `  T
) )
31 ffn 5550 . . . . . . 7  |-  ( h : B --> ( Base `  T )  ->  h  Fn  B )
3229, 30, 313syl 19 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  Fn  B )
33 simpll 731 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  X  C_  B
)
34 fnreseql 5799 . . . . . 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 1184 . . . . 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 5797 . . . . . 6  |-  ( ( g  Fn  B  /\  h  Fn  B )  ->  ( g  =  h  <->  dom  ( g  i^i  h
)  =  B ) )
3710, 32, 36syl2anc 643 . . . . 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 260 . . . 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 30 . . 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 2758 . 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 5099 . . . 4  |-  ( g  =  h  ->  (
g  |`  X )  =  ( h  |`  X ) )
4241eqeq1d 2412 . . 3  |-  ( g  =  h  ->  (
( g  |`  X )  =  F  <->  ( h  |`  X )  =  F ) )
4342rmo4 3087 . 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 204 1  |-  ( ( X  C_  B  /\  ( K `  X )  =  B )  ->  E* g  e.  ( S LMHom  T ) ( g  |`  X )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E*wrmo 2669    i^i cin 3279    C_ wss 3280   dom cdm 4837    |` cres 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002   LMHom clmhm 16050
This theorem is referenced by:  frlmup4  27121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-ghm 14959  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lmhm 16053
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