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Theorem lmhmlsp 17256
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 2454 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 17241 . . . . 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 5672 . . . 4  |-  ( F : V --> ( Base `  T )  ->  Fun  F )
64, 5syl 16 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  Fun  F )
7 lmhmlmod1 17240 . . . . 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 17239 . . . . . . 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 5291 . . . . . . 7  |-  ( F
" U )  C_  ran  F
12 frn 5676 . . . . . . . 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 3478 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( F " U )  C_  ( Base `  T )
)
15 eqid 2454 . . . . . . 7  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
16 lmhmlsp.l . . . . . . 7  |-  L  =  ( LSpan `  T )
172, 15, 16lspcl 17183 . . . . . 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 2454 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
2019, 15lmhmpreima 17255 . . . . 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 3654 . . . . . . 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 5674 . . . . . . . . . 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 3504 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_ 
dom  F )
27 df-ss 3453 . . . . . . . 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 2508 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  =  ( dom  F  i^i  U ) )
30 dminss 5362 . . . . . 6  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
3129, 30syl6eqss 3517 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( F " U
) ) )
322, 16lspssid 17192 . . . . . . 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 5315 . . . . . 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 3477 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  U  C_  ( `' F "
( L `  ( F " U ) ) ) )
37 lmhmlsp.k . . . . 5  |-  K  =  ( LSpan `  S )
3819, 37lspssp 17195 . . . 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 1219 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( K `  U )  C_  ( `' F "
( L `  ( F " U ) ) ) )
40 funimass2 5603 . . 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 17183 . . . . 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 17254 . . . 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 17192 . . . . 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 5315 . . . 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 17195 . . 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 1219 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  C_  V )  ->  ( L `  ( F " U ) )  C_  ( F " ( K `
 U ) ) )
5241, 51eqssd 3484 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 1370    e. wcel 1758    i^i cin 3438    C_ wss 3439   `'ccnv 4950   dom cdm 4951   ran crn 4952   "cima 4954   Fun wfun 5523   -->wf 5525   ` cfv 5529  (class class class)co 6203   Basecbs 14295   LModclmod 17074   LSubSpclss 17139   LSpanclspn 17178   LMHom clmhm 17226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-0g 14502  df-mnd 15537  df-grp 15667  df-minusg 15668  df-sbg 15669  df-subg 15800  df-ghm 15867  df-mgp 16717  df-ur 16729  df-rng 16773  df-lmod 17076  df-lss 17140  df-lsp 17179  df-lmhm 17229
This theorem is referenced by:  frlmup3  18356  lindfmm  18384  lmimlbs  18393  lmhmfgima  29605  lmhmfgsplit  29607
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