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Theorem lmhmlsp 17566
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 2467 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 17551 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : V
--> ( Base `  T
) )
43adantr 465 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  F : V --> ( Base `  T
) )
5 ffun 5739 . . . 4  |-  ( F : V --> ( Base `  T )  ->  Fun  F )
64, 5syl 16 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  Fun  F )
7 lmhmlmod1 17550 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantr 465 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  S  e.  LMod )
9 lmhmlmod2 17549 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
109adantr 465 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  T  e.  LMod )
11 imassrn 5354 . . . . . . 7  |-  ( F
" U )  C_  ran  F
12 frn 5743 . . . . . . . 8  |-  ( F : V --> ( Base `  T )  ->  ran  F 
C_  ( Base `  T
) )
134, 12syl 16 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ran  F 
C_  ( Base `  T
) )
1411, 13syl5ss 3520 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( Base `  T )
)
15 eqid 2467 . . . . . . 7  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
16 lmhmlsp.l . . . . . . 7  |-  L  =  ( LSpan `  T )
172, 15, 16lspcl 17493 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( L `  ( F " U
) )  e.  (
LSubSp `  T ) )
1810, 14, 17syl2anc 661 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)
19 eqid 2467 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019, 15lmhmpreima 17565 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)  ->  ( `' F " ( L `  ( F " U ) ) )  e.  (
LSubSp `  S ) )
2118, 20syldan 470 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( `' F " ( L `
 ( F " U ) ) )  e.  ( LSubSp `  S
) )
22 incom 3696 . . . . . . 7  |-  ( dom 
F  i^i  U )  =  ( U  i^i  dom 
F )
23 simpr 461 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  V )
24 fdm 5741 . . . . . . . . . 10  |-  ( F : V --> ( Base `  T )  ->  dom  F  =  V )
254, 24syl 16 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  dom  F  =  V )
2623, 25sseqtr4d 3546 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_ 
dom  F )
27 df-ss 3495 . . . . . . . 8  |-  ( U 
C_  dom  F  <->  ( U  i^i  dom  F )  =  U )
2826, 27sylib 196 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( U  i^i  dom  F )  =  U )
2922, 28syl5req 2521 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  =  ( dom  F  i^i  U ) )
30 dminss 5426 . . . . . 6  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
3129, 30syl6eqss 3559 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( F " U
) ) )
322, 16lspssid 17502 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( F " U )  C_  ( L `  ( F " U ) ) )
3310, 14, 32syl2anc 661 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( L `  ( F
" U ) ) )
34 imass2 5378 . . . . . 6  |-  ( ( F " U ) 
C_  ( L `  ( F " U ) )  ->  ( `' F " ( F " U ) )  C_  ( `' F " ( L `
 ( F " U ) ) ) )
3533, 34syl 16 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( `' F " ( F
" U ) ) 
C_  ( `' F " ( L `  ( F " U ) ) ) )
3631, 35sstrd 3519 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( L `  ( F " U ) ) ) )
37 lmhmlsp.k . . . . 5  |-  K  =  ( LSpan `  S )
3819, 37lspssp 17505 . . . 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 1228 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )
40 funimass2 5668 . . 3  |-  ( ( Fun  F  /\  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
416, 39, 40syl2anc 661 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
421, 19, 37lspcl 17493 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
438, 23, 42syl2anc 661 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
4419, 15lmhmima 17564 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( K `  U )  e.  ( LSubSp `  S )
)  ->  ( F " ( K `  U
) )  e.  (
LSubSp `  T ) )
4543, 44syldan 470 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  e.  ( LSubSp `  T )
)
461, 37lspssid 17502 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
478, 23, 46syl2anc 661 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
48 imass2 5378 . . . 4  |-  ( U 
C_  ( K `  U )  ->  ( F " U )  C_  ( F " ( K `
 U ) ) )
4947, 48syl 16 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( F " ( K `
 U ) ) )
5015, 16lspssp 17505 . . 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 1228 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  C_  ( F " ( K `
 U ) ) )
5241, 51eqssd 3526 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 369    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   `'ccnv 5004   dom cdm 5005   ran crn 5006   "cima 5008   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507   LModclmod 17383   LSubSpclss 17449   LSpanclspn 17488   LMHom clmhm 17536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lmhm 17539
This theorem is referenced by:  frlmup3  18703  lindfmm  18731  lmimlbs  18740  lmhmfgima  30958  lmhmfgsplit  30960
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