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Theorem lspextmo 17828
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 2485 . . . 4  |-  ( ( ( g  |`  X )  =  F  /\  (
h  |`  X )  =  F )  ->  (
g  |`  X )  =  ( h  |`  X ) )
2 inss1 3714 . . . . . . . . 9  |-  ( g  i^i  h )  C_  g
3 dmss 5212 . . . . . . . . 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 2457 . . . . . . . . . . . . 13  |-  ( Base `  T )  =  (
Base `  T )
75, 6lmhmf 17806 . . . . . . . . . . . 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 5737 . . . . . . . . . . 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 5686 . . . . . . . . 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 3547 . . . . . . 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 755 . . . . . . . 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 17805 . . . . . . . . . . 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 2457 . . . . . . . . . . 11  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019lmhmeql 17827 . . . . . . . . . 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 757 . . . . . . . . 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 17760 . . . . . . . . 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 1228 . . . . . . . 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 3534 . . . . . . 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 3516 . . . . . 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 757 . . . . . . 7  |-  ( ( ( X  C_  B  /\  ( K `  X
)  =  B )  /\  ( g  e.  ( S LMHom  T )  /\  h  e.  ( S LMHom  T ) ) )  ->  h  e.  ( S LMHom  T ) )
305, 6lmhmf 17806 . . . . . . 7  |-  ( h  e.  ( S LMHom  T
)  ->  h : B
--> ( Base `  T
) )
31 ffn 5737 . . . . . . 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 5998 . . . . . 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 1228 . . . . 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 5996 . . . . . 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 2878 . 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 5277 . . . 4  |-  ( g  =  h  ->  (
g  |`  X )  =  ( h  |`  X ) )
4241eqeq1d 2459 . . 3  |-  ( g  =  h  ->  (
( g  |`  X )  =  F  <->  ( h  |`  X )  =  F ) )
4342rmo4 3292 . 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 1395    e. wcel 1819   A.wral 2807   E*wrmo 2810    i^i cin 3470    C_ wss 3471   dom cdm 5008    |` cres 5010    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14643   LModclmod 17638   LSubSpclss 17704   LSpanclspn 17743   LMHom clmhm 17791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-ghm 16391  df-mgp 17268  df-ur 17280  df-ring 17326  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lmhm 17794
This theorem is referenced by:  frlmup4  18961
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