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Theorem lmhmlsp 18272
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 2451 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 18257 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : V
--> ( Base `  T
) )
43adantr 467 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  F : V --> ( Base `  T
) )
5 ffun 5731 . . . 4  |-  ( F : V --> ( Base `  T )  ->  Fun  F )
64, 5syl 17 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  Fun  F )
7 lmhmlmod1 18256 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantr 467 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  S  e.  LMod )
9 lmhmlmod2 18255 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
109adantr 467 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  T  e.  LMod )
11 imassrn 5179 . . . . . . 7  |-  ( F
" U )  C_  ran  F
12 frn 5735 . . . . . . . 8  |-  ( F : V --> ( Base `  T )  ->  ran  F 
C_  ( Base `  T
) )
134, 12syl 17 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ran  F 
C_  ( Base `  T
) )
1411, 13syl5ss 3443 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( Base `  T )
)
15 eqid 2451 . . . . . . 7  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
16 lmhmlsp.l . . . . . . 7  |-  L  =  ( LSpan `  T )
172, 15, 16lspcl 18199 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( L `  ( F " U
) )  e.  (
LSubSp `  T ) )
1810, 14, 17syl2anc 667 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)
19 eqid 2451 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019, 15lmhmpreima 18271 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)  ->  ( `' F " ( L `  ( F " U ) ) )  e.  (
LSubSp `  S ) )
2118, 20syldan 473 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( `' F " ( L `
 ( F " U ) ) )  e.  ( LSubSp `  S
) )
22 incom 3625 . . . . . . 7  |-  ( dom 
F  i^i  U )  =  ( U  i^i  dom 
F )
23 simpr 463 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  V )
24 fdm 5733 . . . . . . . . . 10  |-  ( F : V --> ( Base `  T )  ->  dom  F  =  V )
254, 24syl 17 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  dom  F  =  V )
2623, 25sseqtr4d 3469 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_ 
dom  F )
27 df-ss 3418 . . . . . . . 8  |-  ( U 
C_  dom  F  <->  ( U  i^i  dom  F )  =  U )
2826, 27sylib 200 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( U  i^i  dom  F )  =  U )
2922, 28syl5req 2498 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  =  ( dom  F  i^i  U ) )
30 dminss 5250 . . . . . 6  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
3129, 30syl6eqss 3482 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( F " U
) ) )
322, 16lspssid 18208 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( F " U )  C_  ( L `  ( F " U ) ) )
3310, 14, 32syl2anc 667 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( L `  ( F
" U ) ) )
34 imass2 5204 . . . . . 6  |-  ( ( F " U ) 
C_  ( L `  ( F " U ) )  ->  ( `' F " ( F " U ) )  C_  ( `' F " ( L `
 ( F " U ) ) ) )
3533, 34syl 17 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( `' F " ( F
" U ) ) 
C_  ( `' F " ( L `  ( F " U ) ) ) )
3631, 35sstrd 3442 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( L `  ( F " U ) ) ) )
37 lmhmlsp.k . . . . 5  |-  K  =  ( LSpan `  S )
3819, 37lspssp 18211 . . . 4  |-  ( ( S  e.  LMod  /\  ( `' F " ( L `
 ( F " U ) ) )  e.  ( LSubSp `  S
)  /\  U  C_  ( `' F " ( L `
 ( F " U ) ) ) )  ->  ( K `  U )  C_  ( `' F " ( L `
 ( F " U ) ) ) )
398, 21, 36, 38syl3anc 1268 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )
40 funimass2 5657 . . 3  |-  ( ( Fun  F  /\  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
416, 39, 40syl2anc 667 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
421, 19, 37lspcl 18199 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
438, 23, 42syl2anc 667 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
4419, 15lmhmima 18270 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( K `  U )  e.  ( LSubSp `  S )
)  ->  ( F " ( K `  U
) )  e.  (
LSubSp `  T ) )
4543, 44syldan 473 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  e.  ( LSubSp `  T )
)
461, 37lspssid 18208 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
478, 23, 46syl2anc 667 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
48 imass2 5204 . . . 4  |-  ( U 
C_  ( K `  U )  ->  ( F " U )  C_  ( F " ( K `
 U ) ) )
4947, 48syl 17 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( F " ( K `
 U ) ) )
5015, 16lspssp 18211 . . 3  |-  ( ( T  e.  LMod  /\  ( F " ( K `  U ) )  e.  ( LSubSp `  T )  /\  ( F " U
)  C_  ( F " ( K `  U
) ) )  -> 
( L `  ( F " U ) ) 
C_  ( F "
( K `  U
) ) )
5110, 45, 49, 50syl3anc 1268 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  C_  ( F " ( K `
 U ) ) )
5241, 51eqssd 3449 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  =  ( L `  ( F " U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    i^i cin 3403    C_ wss 3404   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   Basecbs 15121   LModclmod 18091   LSubSpclss 18155   LSpanclspn 18194   LMHom clmhm 18242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-sbg 16675  df-subg 16814  df-ghm 16881  df-mgp 17724  df-ur 17736  df-ring 17782  df-lmod 18093  df-lss 18156  df-lsp 18195  df-lmhm 18245
This theorem is referenced by:  frlmup3  19358  lindfmm  19385  lmimlbs  19394  lmhmfgima  35942  lmhmfgsplit  35944
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