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

Theorem lspextmo 17137
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 2462 . . . 4  |-  ( ( ( g  |`  X )  =  F  /\  (
h  |`  X )  =  F )  ->  (
g  |`  X )  =  ( h  |`  X ) )
2 inss1 3570 . . . . . . . . 9  |-  ( g  i^i  h )  C_  g
3 dmss 5039 . . . . . . . . 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 2443 . . . . . . . . . . . . 13  |-  ( Base `  T )  =  (
Base `  T )
75, 6lmhmf 17115 . . . . . . . . . . . 12  |-  ( g  e.  ( S LMHom  T
)  ->  g : B
--> ( Base `  T
) )
87ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  g : B
--> ( Base `  T
) )
9 ffn 5559 . . . . . . . . . . 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 716 . . . . . . . . 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 5510 . . . . . . . . 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 3404 . . . . . . 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 754 . . . . . . . 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 17114 . . . . . . . . . . 11  |-  ( g  e.  ( S LMHom  T
)  ->  S  e.  LMod )
1716adantr 465 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
1817ad2antrl 727 . . . . . . . . 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 2443 . . . . . . . . . . 11  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019lmhmeql 17136 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  dom  ( g  i^i  h )  e.  ( LSubSp `  S )
)
2120ad2antrl 727 . . . . . . . . 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 756 . . . . . . . . 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 17069 . . . . . . . . 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 1218 . . . . . . . 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 3391 . . . . . . 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 3373 . . . . . 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 615 . . . . 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 756 . . . . . . 7  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  e.  ( S LMHom  T ) )
305, 6lmhmf 17115 . . . . . . 7  |-  ( h  e.  ( S LMHom  T
)  ->  h : B
--> ( Base `  T
) )
31 ffn 5559 . . . . . . 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 753 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  X  C_  B
)
34 fnreseql 5813 . . . . . 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 1218 . . . . 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 5811 . . . . . 6  |-  ( ( g  Fn  B  /\  h  Fn  B )  ->  ( g  =  h  <->  dom  ( g  i^i  h
)  =  B ) )
3710, 32, 36syl2anc 661 . . . . 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 2808 . 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 5104 . . . 4  |-  ( g  =  h  ->  (
g  |`  X )  =  ( h  |`  X ) )
4241eqeq1d 2451 . . 3  |-  ( g  =  h  ->  (
( g  |`  X )  =  F  <->  ( h  |`  X )  =  F ) )
4342rmo4 3152 . 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 1369    e. wcel 1756   A.wral 2715   E*wrmo 2718    i^i cin 3327    C_ wss 3328   dom cdm 4840    |` cres 4842    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   Basecbs 14174   LModclmod 16948   LSubSpclss 17013   LSpanclspn 17052   LMHom clmhm 17100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-ghm 15745  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lmhm 17103
This theorem is referenced by:  frlmup4  18229
  Copyright terms: Public domain W3C validator