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Theorem pjfo 18260
Description: A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjf.k  |-  K  =  ( proj `  W
)
pjf.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
pjfo  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( K `  T
) : V -onto-> T
)

Proof of Theorem pjfo
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pjf.k . . 3  |-  K  =  ( proj `  W
)
2 pjf.v . . 3  |-  V  =  ( Base `  W
)
31, 2pjf2 18259 . 2  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( K `  T
) : V --> T )
4 frn 5668 . . . 4  |-  ( ( K `  T ) : V --> T  ->  ran  ( K `  T
)  C_  T )
53, 4syl 16 . . 3  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  ran  ( K `  T
)  C_  T )
6 eqid 2452 . . . . . . . . . 10  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2452 . . . . . . . . . 10  |-  ( proj1 `  W )  =  ( proj1 `  W )
86, 7, 1pjval 18255 . . . . . . . . 9  |-  ( T  e.  dom  K  -> 
( K `  T
)  =  ( T ( proj1 `  W ) ( ( ocv `  W ) `
 T ) ) )
98ad2antlr 726 . . . . . . . 8  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  ( K `  T )  =  ( T (
proj1 `  W
) ( ( ocv `  W ) `  T
) ) )
109fveq1d 5796 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( K `  T
) `  x )  =  ( ( T ( proj1 `  W ) ( ( ocv `  W ) `
 T ) ) `
 x ) )
11 eqid 2452 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
12 eqid 2452 . . . . . . . 8  |-  ( LSSum `  W )  =  (
LSSum `  W )
13 eqid 2452 . . . . . . . 8  |-  ( 0g
`  W )  =  ( 0g `  W
)
14 eqid 2452 . . . . . . . 8  |-  (Cntz `  W )  =  (Cntz `  W )
15 phllmod 18179 . . . . . . . . . . 11  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1615adantr 465 . . . . . . . . . 10  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  W  e.  LMod )
17 eqid 2452 . . . . . . . . . . 11  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1817lsssssubg 17157 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
1916, 18syl 16 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( LSubSp `  W )  C_  (SubGrp `  W )
)
202, 17, 6, 12, 1pjdm2 18256 . . . . . . . . . 10  |-  ( W  e.  PreHil  ->  ( T  e. 
dom  K  <->  ( T  e.  ( LSubSp `  W )  /\  ( T ( LSSum `  W ) ( ( ocv `  W ) `
 T ) )  =  V ) ) )
2120simprbda 623 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  e.  ( LSubSp `  W ) )
2219, 21sseldd 3460 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  e.  (SubGrp `  W
) )
232, 17lssss 17136 . . . . . . . . . . 11  |-  ( T  e.  ( LSubSp `  W
)  ->  T  C_  V
)
2421, 23syl 16 . . . . . . . . . 10  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  C_  V )
252, 6, 17ocvlss 18217 . . . . . . . . . 10  |-  ( ( W  e.  PreHil  /\  T  C_  V )  ->  (
( ocv `  W
) `  T )  e.  ( LSubSp `  W )
)
2624, 25syldan 470 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( ( ocv `  W
) `  T )  e.  ( LSubSp `  W )
)
2719, 26sseldd 3460 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( ( ocv `  W
) `  T )  e.  (SubGrp `  W )
)
286, 17, 13ocvin 18219 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  ( LSubSp `  W )
)  ->  ( T  i^i  ( ( ocv `  W
) `  T )
)  =  { ( 0g `  W ) } )
2921, 28syldan 470 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( T  i^i  (
( ocv `  W
) `  T )
)  =  { ( 0g `  W ) } )
30 lmodabl 17110 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Abel )
3116, 30syl 16 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  W  e.  Abel )
3214, 31, 22, 27ablcntzd 16455 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  C_  ( (Cntz `  W ) `  (
( ocv `  W
) `  T )
) )
3311, 12, 13, 14, 22, 27, 29, 32, 7pj1lid 16314 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( T ( proj1 `  W )
( ( ocv `  W
) `  T )
) `  x )  =  x )
3410, 33eqtrd 2493 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( K `  T
) `  x )  =  x )
35 ffn 5662 . . . . . . . . 9  |-  ( ( K `  T ) : V --> T  -> 
( K `  T
)  Fn  V )
363, 35syl 16 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( K `  T
)  Fn  V )
3736adantr 465 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  ( K `  T )  Fn  V )
3824sselda 3459 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  x  e.  V )
39 fnfvelrn 5944 . . . . . . 7  |-  ( ( ( K `  T
)  Fn  V  /\  x  e.  V )  ->  ( ( K `  T ) `  x
)  e.  ran  ( K `  T )
)
4037, 38, 39syl2anc 661 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( K `  T
) `  x )  e.  ran  ( K `  T ) )
4134, 40eqeltrrd 2541 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  x  e.  ran  ( K `  T ) )
4241ex 434 . . . 4  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( x  e.  T  ->  x  e.  ran  ( K `  T )
) )
4342ssrdv 3465 . . 3  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  C_  ran  ( K `
 T ) )
445, 43eqssd 3476 . 2  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  ran  ( K `  T
)  =  T )
45 dffo2 5727 . 2  |-  ( ( K `  T ) : V -onto-> T  <->  ( ( K `  T ) : V --> T  /\  ran  ( K `  T )  =  T ) )
463, 44, 45sylanbrc 664 1  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( K `  T
) : V -onto-> T
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    i^i cin 3430    C_ wss 3431   {csn 3980   dom cdm 4943   ran crn 4944    Fn wfn 5516   -->wf 5517   -onto->wfo 5519   ` cfv 5521  (class class class)co 6195   Basecbs 14287   +g cplusg 14352   0gc0g 14492  SubGrpcsubg 15789  Cntzccntz 15947   LSSumclsm 16249   proj1cpj1 16250   Abelcabel 16394   LModclmod 17066   LSubSpclss 17131   PreHilcphl 18173   ocvcocv 18205   projcpj 18245
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-sca 14368  df-vsca 14369  df-ip 14370  df-0g 14494  df-mnd 15529  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-subg 15792  df-ghm 15859  df-cntz 15949  df-lsm 16251  df-pj1 16252  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-lmod 17068  df-lss 17132  df-lmhm 17221  df-lvec 17302  df-sra 17371  df-rgmod 17372  df-phl 18175  df-ocv 18208  df-pj 18248
This theorem is referenced by: (None)
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