Theorem pjfo 20059

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.

Step	Hyp	Ref	Expression
1		pjf.k	. . 3 |- K = ( proj ` W )
2		pjf.v	. . 3 |- V = ( Base ` W )
3	1, 2	pjf2 20058	. 2 |- ( ( W e. PreHil /\ T e. dom K ) -> ( K ` T ) : V --> T )
4		frn 6053	. . . 4 |- ( ( K ` T ) : V --> T -> ran ( K ` T ) C_ T )
5	3, 4	syl 17	. . 3 |- ( ( W e. PreHil /\ T e. dom K ) -> ran ( K ` T ) C_ T )
6		eqid 2622	. . . . . . . . . 10 |- ( ocv ` W ) = ( ocv ` W )
7		eqid 2622	. . . . . . . . . 10 |- ( proj1 ` W ) = ( proj1 ` W )
8	6, 7, 1	pjval 20054	. . . . . . . . 9 |- ( T e. dom K -> ( K ` T ) = ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) )
9	8	ad2antlr 763	. . . . . . . 8 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> ( K ` T ) = ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) )
10	9	fveq1d 6193	. . . . . . 7 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> ( ( K ` T ) ` x ) = ( ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) ` x ) )
11		eqid 2622	. . . . . . . 8 |- ( +g ` W ) = ( +g ` W )
12		eqid 2622	. . . . . . . 8 |- ( LSSum ` W ) = ( LSSum ` W )
13		eqid 2622	. . . . . . . 8 |- ( 0g ` W ) = ( 0g ` W )
14		eqid 2622	. . . . . . . 8 |- (Cntz ` W ) = (Cntz ` W )
15		phllmod 19975	. . . . . . . . . . 11 |- ( W e. PreHil -> W e. LMod )
16	15	adantr 481	. . . . . . . . . 10 |- ( ( W e. PreHil /\ T e. dom K ) -> W e. LMod )
17		eqid 2622	. . . . . . . . . . 11 |- ( LSubSp ` W ) = ( LSubSp ` W )
18	17	lsssssubg 18958	. . . . . . . . . 10 |- ( W e. LMod -> ( LSubSp ` W ) C_ (SubGrp ` W ) )
19	16, 18	syl 17	. . . . . . . . 9 |- ( ( W e. PreHil /\ T e. dom K ) -> ( LSubSp ` W ) C_ (SubGrp ` W ) )
20	2, 17, 6, 12, 1	pjdm2 20055	. . . . . . . . . 10 |- ( W e. PreHil -> ( T e. dom K <-> ( T e. ( LSubSp ` W ) /\ ( T ( LSSum ` W ) ( ( ocv ` W ) ` T ) ) = V ) ) )
21	20	simprbda 653	. . . . . . . . 9 |- ( ( W e. PreHil /\ T e. dom K ) -> T e. ( LSubSp ` W ) )
22	19, 21	sseldd 3604	. . . . . . . 8 |- ( ( W e. PreHil /\ T e. dom K ) -> T e. (SubGrp ` W ) )
23	2, 17	lssss 18937	. . . . . . . . . . 11 |- ( T e. ( LSubSp ` W ) -> T C_ V )
24	21, 23	syl 17	. . . . . . . . . 10 |- ( ( W e. PreHil /\ T e. dom K ) -> T C_ V )
25	2, 6, 17	ocvlss 20016	. . . . . . . . . 10 |- ( ( W e. PreHil /\ T C_ V ) -> ( ( ocv ` W ) ` T ) e. ( LSubSp ` W ) )
26	24, 25	syldan 487	. . . . . . . . 9 |- ( ( W e. PreHil /\ T e. dom K ) -> ( ( ocv ` W ) ` T ) e. ( LSubSp ` W ) )
27	19, 26	sseldd 3604	. . . . . . . 8 |- ( ( W e. PreHil /\ T e. dom K ) -> ( ( ocv ` W ) ` T ) e. (SubGrp ` W ) )
28	6, 17, 13	ocvin 20018	. . . . . . . . 9 |- ( ( W e. PreHil /\ T e. ( LSubSp ` W ) ) -> ( T i^i ( ( ocv ` W ) ` T ) ) = { ( 0g ` W ) } )
29	21, 28	syldan 487	. . . . . . . 8 |- ( ( W e. PreHil /\ T e. dom K ) -> ( T i^i ( ( ocv ` W ) ` T ) ) = { ( 0g ` W ) } )
30		lmodabl 18910	. . . . . . . . . 10 |- ( W e. LMod -> W e. Abel )
31	16, 30	syl 17	. . . . . . . . 9 |- ( ( W e. PreHil /\ T e. dom K ) -> W e. Abel )
32	14, 31, 22, 27	ablcntzd 18260	. . . . . . . 8 |- ( ( W e. PreHil /\ T e. dom K ) -> T C_ ( (Cntz ` W ) ` ( ( ocv ` W ) ` T ) ) )
33	11, 12, 13, 14, 22, 27, 29, 32, 7	pj1lid 18114	. . . . . . 7 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> ( ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) ` x ) = x )
34	10, 33	eqtrd 2656	. . . . . 6 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> ( ( K ` T ) ` x ) = x )
35		ffn 6045	. . . . . . . . 9 |- ( ( K ` T ) : V --> T -> ( K ` T ) Fn V )
36	3, 35	syl 17	. . . . . . . 8 |- ( ( W e. PreHil /\ T e. dom K ) -> ( K ` T ) Fn V )
37	36	adantr 481	. . . . . . 7 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> ( K ` T ) Fn V )
38	24	sselda 3603	. . . . . . 7 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> x e. V )
39		fnfvelrn 6356	. . . . . . 7 |- ( ( ( K ` T ) Fn V /\ x e. V ) -> ( ( K ` T ) ` x ) e. ran ( K ` T ) )
40	37, 38, 39	syl2anc 693	. . . . . 6 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> ( ( K ` T ) ` x ) e. ran ( K ` T ) )
41	34, 40	eqeltrrd 2702	. . . . 5 |- ( ( ( W e. PreHil /\ T e. dom K ) /\ x e. T ) -> x e. ran ( K ` T ) )
42	41	ex 450	. . . 4 |- ( ( W e. PreHil /\ T e. dom K ) -> ( x e. T -> x e. ran ( K ` T ) ) )
43	42	ssrdv 3609	. . 3 |- ( ( W e. PreHil /\ T e. dom K ) -> T C_ ran ( K ` T ) )
44	5, 43	eqssd 3620	. 2 |- ( ( W e. PreHil /\ T e. dom K ) -> ran ( K ` T ) = T )
45		dffo2 6119	. 2 |- ( ( K ` T ) : V -onto-> T <-> ( ( K ` T ) : V --> T /\ ran ( K ` T ) = T ) )
46	3, 44, 45	sylanbrc 698	1 |- ( ( W e. PreHil /\ T e. dom K ) -> ( K ` T ) : V -onto-> T )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   {csn 4177   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100  SubGrpcsubg 17588  Cntzccntz 17748   LSSumclsm 18049   proj1cpj1 18050   Abelcabl 18194   LModclmod 18863   LSubSpclss 18932   PreHilcphl 19969   ocvcocv 20004   projcpj 20044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971  df-ocv 20007  df-pj 20047

This theorem is referenced by: (None)