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Theorem lmhmlsp 15822
Description: Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlsp.v  |-  V  =  ( Base `  S
)
lmhmlsp.k  |-  K  =  ( LSpan `  S )
lmhmlsp.l  |-  L  =  ( LSpan `  T )
Assertion
Ref Expression
lmhmlsp  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  =  ( L `  ( F " U ) ) )

Proof of Theorem lmhmlsp
StepHypRef Expression
1 lmhmlsp.v . . . . . 6  |-  V  =  ( Base `  S
)
2 eqid 2296 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 15807 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : V
--> ( Base `  T
) )
43adantr 451 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  F : V --> ( Base `  T
) )
5 ffun 5407 . . . 4  |-  ( F : V --> ( Base `  T )  ->  Fun  F )
64, 5syl 15 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  Fun  F )
7 lmhmlmod1 15806 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantr 451 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  S  e.  LMod )
9 lmhmlmod2 15805 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
109adantr 451 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  T  e.  LMod )
11 imassrn 5041 . . . . . . 7  |-  ( F
" U )  C_  ran  F
12 frn 5411 . . . . . . . 8  |-  ( F : V --> ( Base `  T )  ->  ran  F 
C_  ( Base `  T
) )
134, 12syl 15 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ran  F 
C_  ( Base `  T
) )
1411, 13syl5ss 3203 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( Base `  T )
)
15 eqid 2296 . . . . . . 7  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
16 lmhmlsp.l . . . . . . 7  |-  L  =  ( LSpan `  T )
172, 15, 16lspcl 15749 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( L `  ( F " U
) )  e.  (
LSubSp `  T ) )
1810, 14, 17syl2anc 642 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)
19 eqid 2296 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019, 15lmhmpreima 15821 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)  ->  ( `' F " ( L `  ( F " U ) ) )  e.  (
LSubSp `  S ) )
2118, 20syldan 456 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( `' F " ( L `
 ( F " U ) ) )  e.  ( LSubSp `  S
) )
22 incom 3374 . . . . . . 7  |-  ( dom 
F  i^i  U )  =  ( U  i^i  dom 
F )
23 simpr 447 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  V )
24 fdm 5409 . . . . . . . . . 10  |-  ( F : V --> ( Base `  T )  ->  dom  F  =  V )
254, 24syl 15 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  dom  F  =  V )
2623, 25sseqtr4d 3228 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_ 
dom  F )
27 df-ss 3179 . . . . . . . 8  |-  ( U 
C_  dom  F  <->  ( U  i^i  dom  F )  =  U )
2826, 27sylib 188 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( U  i^i  dom  F )  =  U )
2922, 28syl5req 2341 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  =  ( dom  F  i^i  U ) )
30 dminss 5111 . . . . . . 7  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
3130a1i 10 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( dom  F  i^i  U ) 
C_  ( `' F " ( F " U
) ) )
3229, 31eqsstrd 3225 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( F " U
) ) )
332, 16lspssid 15758 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( F " U )  C_  ( L `  ( F " U ) ) )
3410, 14, 33syl2anc 642 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( L `  ( F
" U ) ) )
35 imass2 5065 . . . . . 6  |-  ( ( F " U ) 
C_  ( L `  ( F " U ) )  ->  ( `' F " ( F " U ) )  C_  ( `' F " ( L `
 ( F " U ) ) ) )
3634, 35syl 15 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( `' F " ( F
" U ) ) 
C_  ( `' F " ( L `  ( F " U ) ) ) )
3732, 36sstrd 3202 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( L `  ( F " U ) ) ) )
38 lmhmlsp.k . . . . 5  |-  K  =  ( LSpan `  S )
3919, 38lspssp 15761 . . . 4  |-  ( ( S  e.  LMod  /\  ( `' F " ( L `
 ( F " U ) ) )  e.  ( LSubSp `  S
)  /\  U  C_  ( `' F " ( L `
 ( F " U ) ) ) )  ->  ( K `  U )  C_  ( `' F " ( L `
 ( F " U ) ) ) )
408, 21, 37, 39syl3anc 1182 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )
41 funimass2 5342 . . 3  |-  ( ( Fun  F  /\  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
426, 40, 41syl2anc 642 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
431, 19, 38lspcl 15749 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
448, 23, 43syl2anc 642 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
4519, 15lmhmima 15820 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( K `  U )  e.  ( LSubSp `  S )
)  ->  ( F " ( K `  U
) )  e.  (
LSubSp `  T ) )
4644, 45syldan 456 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  e.  ( LSubSp `  T )
)
471, 38lspssid 15758 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
488, 23, 47syl2anc 642 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
49 imass2 5065 . . . 4  |-  ( U 
C_  ( K `  U )  ->  ( F " U )  C_  ( F " ( K `
 U ) ) )
5048, 49syl 15 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( F " ( K `
 U ) ) )
5115, 16lspssp 15761 . . 3  |-  ( ( T  e.  LMod  /\  ( F " ( K `  U ) )  e.  ( LSubSp `  T )  /\  ( F " U
)  C_  ( F " ( K `  U
) ) )  -> 
( L `  ( F " U ) ) 
C_  ( F "
( K `  U
) ) )
5210, 46, 50, 51syl3anc 1182 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  C_  ( F " ( K `
 U ) ) )
5342, 52eqssd 3209 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  =  ( L `  ( F " U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LMHom clmhm 15792
This theorem is referenced by:  lmhmfgima  27285  lmhmfgsplit  27287  frlmup3  27355  lindfmm  27400  lmimlbs  27409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lmhm 15795
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