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Theorem pjfo 16615
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 16614 . 2  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( K `  T
) : V --> T )
4 frn 5395 . . . 4  |-  ( ( K `  T ) : V --> T  ->  ran  ( K `  T
)  C_  T )
53, 4syl 15 . . 3  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  ran  ( K `  T
)  C_  T )
6 eqid 2283 . . . . . . . . . 10  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2283 . . . . . . . . . 10  |-  ( proj
1 `  W )  =  ( proj 1 `  W )
86, 7, 1pjval 16610 . . . . . . . . 9  |-  ( T  e.  dom  K  -> 
( K `  T
)  =  ( T ( proj 1 `  W ) ( ( ocv `  W ) `
 T ) ) )
98ad2antlr 707 . . . . . . . 8  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  ( K `  T )  =  ( T (
proj 1 `  W ) ( ( ocv `  W
) `  T )
) )
109fveq1d 5527 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( K `  T
) `  x )  =  ( ( T ( proj 1 `  W ) ( ( ocv `  W ) `
 T ) ) `
 x ) )
11 eqid 2283 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
12 eqid 2283 . . . . . . . 8  |-  ( LSSum `  W )  =  (
LSSum `  W )
13 eqid 2283 . . . . . . . 8  |-  ( 0g
`  W )  =  ( 0g `  W
)
14 eqid 2283 . . . . . . . 8  |-  (Cntz `  W )  =  (Cntz `  W )
15 phllmod 16534 . . . . . . . . . . 11  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1615adantr 451 . . . . . . . . . 10  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  W  e.  LMod )
17 eqid 2283 . . . . . . . . . . 11  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1817lsssssubg 15715 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
1916, 18syl 15 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( LSubSp `  W )  C_  (SubGrp `  W )
)
202, 17, 6, 12, 1pjdm2 16611 . . . . . . . . . 10  |-  ( W  e.  PreHil  ->  ( T  e. 
dom  K  <->  ( T  e.  ( LSubSp `  W )  /\  ( T ( LSSum `  W ) ( ( ocv `  W ) `
 T ) )  =  V ) ) )
2120simprbda 606 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  e.  ( LSubSp `  W ) )
2219, 21sseldd 3181 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  e.  (SubGrp `  W
) )
232, 17lssss 15694 . . . . . . . . . . 11  |-  ( T  e.  ( LSubSp `  W
)  ->  T  C_  V
)
2421, 23syl 15 . . . . . . . . . 10  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  C_  V )
252, 6, 17ocvlss 16572 . . . . . . . . . 10  |-  ( ( W  e.  PreHil  /\  T  C_  V )  ->  (
( ocv `  W
) `  T )  e.  ( LSubSp `  W )
)
2624, 25syldan 456 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( ( ocv `  W
) `  T )  e.  ( LSubSp `  W )
)
2719, 26sseldd 3181 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( ( ocv `  W
) `  T )  e.  (SubGrp `  W )
)
286, 17, 13ocvin 16574 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  ( LSubSp `  W )
)  ->  ( T  i^i  ( ( ocv `  W
) `  T )
)  =  { ( 0g `  W ) } )
2921, 28syldan 456 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( T  i^i  (
( ocv `  W
) `  T )
)  =  { ( 0g `  W ) } )
30 lmodabl 15672 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Abel )
3116, 30syl 15 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  W  e.  Abel )
3214, 31, 22, 27ablcntzd 15149 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  C_  ( (Cntz `  W ) `  (
( ocv `  W
) `  T )
) )
3311, 12, 13, 14, 22, 27, 29, 32, 7pj1lid 15010 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( T ( proj
1 `  W )
( ( ocv `  W
) `  T )
) `  x )  =  x )
3410, 33eqtrd 2315 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( K `  T
) `  x )  =  x )
35 ffn 5389 . . . . . . . . 9  |-  ( ( K `  T ) : V --> T  -> 
( K `  T
)  Fn  V )
363, 35syl 15 . . . . . . . 8  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( K `  T
)  Fn  V )
3736adantr 451 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  ( K `  T )  Fn  V )
3824sselda 3180 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  x  e.  V )
39 fnfvelrn 5662 . . . . . . 7  |-  ( ( ( K `  T
)  Fn  V  /\  x  e.  V )  ->  ( ( K `  T ) `  x
)  e.  ran  ( K `  T )
)
4037, 38, 39syl2anc 642 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  (
( K `  T
) `  x )  e.  ran  ( K `  T ) )
4134, 40eqeltrrd 2358 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  T  e.  dom  K
)  /\  x  e.  T )  ->  x  e.  ran  ( K `  T ) )
4241ex 423 . . . 4  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( x  e.  T  ->  x  e.  ran  ( K `  T )
) )
4342ssrdv 3185 . . 3  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  T  C_  ran  ( K `
 T ) )
445, 43eqssd 3196 . 2  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  ->  ran  ( K `  T
)  =  T )
45 dffo2 5455 . 2  |-  ( ( K `  T ) : V -onto-> T  <->  ( ( K `  T ) : V --> T  /\  ran  ( K `  T )  =  T ) )
463, 44, 45sylanbrc 645 1  |-  ( ( W  e.  PreHil  /\  T  e.  dom  K )  -> 
( K `  T
) : V -onto-> T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   {csn 3640   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400  SubGrpcsubg 14615  Cntzccntz 14791   LSSumclsm 14945   proj
1cpj1 14946   Abelcabel 15090   LModclmod 15627   LSubSpclss 15689   PreHilcphl 16528   ocvcocv 16560   projcpj 16600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-cntz 14793  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lmhm 15779  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-phl 16530  df-ocv 16563  df-pj 16603
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