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Theorem lspextmo 15813
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 2302 . . . 4  |-  ( ( ( g  |`  X )  =  F  /\  (
h  |`  X )  =  F )  ->  (
g  |`  X )  =  ( h  |`  X ) )
2 inss1 3389 . . . . . . . . 9  |-  ( g  i^i  h )  C_  g
3 dmss 4878 . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( Base `  T )  =  (
Base `  T )
75, 6lmhmf 15791 . . . . . . . . . . . 12  |-  ( g  e.  ( S LMHom  T
)  ->  g : B
--> ( Base `  T
) )
87ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  g : B
--> ( Base `  T
) )
9 ffn 5389 . . . . . . . . . . 11  |-  ( g : B --> ( Base `  T )  ->  g  Fn  B )
108, 9syl 15 . . . . . . . . . 10  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  g  Fn  B )
1110adantrr 697 . . . . . . . . 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 5343 . . . . . . . . 9  |-  ( g  Fn  B  ->  dom  g  =  B )
1311, 12syl 15 . . . . . . . 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 3226 . . . . . . 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 731 . . . . . . . 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 15790 . . . . . . . . . . 11  |-  ( g  e.  ( S LMHom  T
)  ->  S  e.  LMod )
1716adantr 451 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
1817ad2antrl 708 . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019lmhmeql 15812 . . . . . . . . . 10  |-  ( ( g  e.  ( S LMHom 
T )  /\  h  e.  ( S LMHom  T ) )  ->  dom  ( g  i^i  h )  e.  ( LSubSp `  S )
)
2120ad2antrl 708 . . . . . . . . 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 733 . . . . . . . . 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 15745 . . . . . . . . 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 1182 . . . . . . . 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 3213 . . . . . . 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 3196 . . . . . 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 598 . . . . 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 733 . . . . . . 7  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  e.  ( S LMHom  T ) )
305, 6lmhmf 15791 . . . . . . 7  |-  ( h  e.  ( S LMHom  T
)  ->  h : B
--> ( Base `  T
) )
31 ffn 5389 . . . . . . 7  |-  ( h : B --> ( Base `  T )  ->  h  Fn  B )
3229, 30, 313syl 18 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  Fn  B )
33 simpll 730 . . . . . 6  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  X  C_  B
)
34 fnreseql 5635 . . . . . 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 1182 . . . . 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 5633 . . . . . 6  |-  ( ( g  Fn  B  /\  h  Fn  B )  ->  ( g  =  h  <->  dom  ( g  i^i  h
)  =  B ) )
3710, 32, 36syl2anc 642 . . . . 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 259 . . . 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 28 . . 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 2635 . 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 4949 . . . 4  |-  ( g  =  h  ->  (
g  |`  X )  =  ( h  |`  X ) )
4241eqeq1d 2291 . . 3  |-  ( g  =  h  ->  (
( g  |`  X )  =  F  <->  ( h  |`  X )  =  F ) )
4342rmo4 2958 . 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 203 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E*wrmo 2546    i^i cin 3151    C_ wss 3152   dom cdm 4689    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LMHom clmhm 15776
This theorem is referenced by:  frlmup4  27253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779
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