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Theorem lmhmlsp 16118
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 2436 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 16103 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F : V
--> ( Base `  T
) )
43adantr 452 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  F : V --> ( Base `  T
) )
5 ffun 5586 . . . 4  |-  ( F : V --> ( Base `  T )  ->  Fun  F )
64, 5syl 16 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  Fun  F )
7 lmhmlmod1 16102 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantr 452 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  S  e.  LMod )
9 lmhmlmod2 16101 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
109adantr 452 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  T  e.  LMod )
11 imassrn 5209 . . . . . . 7  |-  ( F
" U )  C_  ran  F
12 frn 5590 . . . . . . . 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 3352 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( Base `  T )
)
15 eqid 2436 . . . . . . 7  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
16 lmhmlsp.l . . . . . . 7  |-  L  =  ( LSpan `  T )
172, 15, 16lspcl 16045 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( L `  ( F " U
) )  e.  (
LSubSp `  T ) )
1810, 14, 17syl2anc 643 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)
19 eqid 2436 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019, 15lmhmpreima 16117 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( L `  ( F " U ) )  e.  ( LSubSp `  T )
)  ->  ( `' F " ( L `  ( F " U ) ) )  e.  (
LSubSp `  S ) )
2118, 20syldan 457 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( `' F " ( L `
 ( F " U ) ) )  e.  ( LSubSp `  S
) )
22 incom 3526 . . . . . . 7  |-  ( dom 
F  i^i  U )  =  ( U  i^i  dom 
F )
23 simpr 448 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  V )
24 fdm 5588 . . . . . . . . . 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 3378 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_ 
dom  F )
27 df-ss 3327 . . . . . . . 8  |-  ( U 
C_  dom  F  <->  ( U  i^i  dom  F )  =  U )
2826, 27sylib 189 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( U  i^i  dom  F )  =  U )
2922, 28syl5req 2481 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  =  ( dom  F  i^i  U ) )
30 dminss 5279 . . . . . 6  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
3129, 30syl6eqss 3391 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( F " U
) ) )
322, 16lspssid 16054 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " U )  C_  ( Base `  T )
)  ->  ( F " U )  C_  ( L `  ( F " U ) ) )
3310, 14, 32syl2anc 643 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( L `  ( F
" U ) ) )
34 imass2 5233 . . . . . 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 3351 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( L `  ( F " U ) ) ) )
37 lmhmlsp.k . . . . 5  |-  K  =  ( LSpan `  S )
3819, 37lspssp 16057 . . . 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 1184 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )
40 funimass2 5520 . . 3  |-  ( ( Fun  F  /\  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
416, 39, 40syl2anc 643 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  C_  ( L `  ( F
" U ) ) )
421, 19, 37lspcl 16045 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
438, 23, 42syl2anc 643 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  e.  ( LSubSp `  S )
)
4419, 15lmhmima 16116 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  ( K `  U )  e.  ( LSubSp `  S )
)  ->  ( F " ( K `  U
) )  e.  (
LSubSp `  T ) )
4543, 44syldan 457 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  e.  ( LSubSp `  T )
)
461, 37lspssid 16054 . . . . 5  |-  ( ( S  e.  LMod  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
478, 23, 46syl2anc 643 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( K `  U
) )
48 imass2 5233 . . . 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 16057 . . 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 1184 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  C_  ( F " ( K `
 U ) ) )
5241, 51eqssd 3358 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 359    = wceq 1652    e. wcel 1725    i^i cin 3312    C_ wss 3313   `'ccnv 4870   dom cdm 4871   ran crn 4872   "cima 4874   Fun wfun 5441   -->wf 5443   ` cfv 5447  (class class class)co 6074   Basecbs 13462   LModclmod 15943   LSubSpclss 16001   LSpanclspn 16040   LMHom clmhm 16088
This theorem is referenced by:  lmhmfgima  27151  lmhmfgsplit  27153  frlmup3  27221  lindfmm  27266  lmimlbs  27275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-0g 13720  df-mnd 14683  df-grp 14805  df-minusg 14806  df-sbg 14807  df-subg 14934  df-ghm 14997  df-mgp 15642  df-rng 15656  df-ur 15658  df-lmod 15945  df-lss 16002  df-lsp 16041  df-lmhm 16091
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