Theorem pjfval 20050

Description: The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypotheses

Ref	Expression
pjfval.v	|- V = ( Base ` W )
pjfval.l	|- L = ( LSubSp ` W )
pjfval.o	|- ._|_ = ( ocv ` W )
pjfval.p	|- P = ( proj1 ` W )
pjfval.k	|- K = ( proj ` W )

Assertion

Ref	Expression
pjfval	|- K = ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) )

Distinct variable groups:   x, ._|_   x, L   x, P   x, V   x, W

Allowed substitution hint:   K( x)

Proof of Theorem pjfval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjfval.k	. 2 |- K = ( proj ` W )
2		fveq2 6191	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
3		pjfval.l	. . . . . . 7 |- L = ( LSubSp ` W )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = L )
5		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( proj1 ` w ) = ( proj1 ` W ) )
6		pjfval.p	. . . . . . . 8 |- P = ( proj1 ` W )
7	5, 6	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( proj1 ` w ) = P )
8		eqidd 2623	. . . . . . 7 |- ( w = W -> x = x )
9		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( ocv ` w ) = ( ocv ` W ) )
10		pjfval.o	. . . . . . . . 9 |- ._|_ = ( ocv ` W )
11	9, 10	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( ocv ` w ) = ._|_ )
12	11	fveq1d 6193	. . . . . . 7 |- ( w = W -> ( ( ocv ` w ) ` x ) = ( ._|_ ` x ) )
13	7, 8, 12	oveq123d 6671	. . . . . 6 |- ( w = W -> ( x ( proj1 ` w ) ( ( ocv ` w ) ` x ) ) = ( x P ( ._|_ ` x ) ) )
14	4, 13	mpteq12dv 4733	. . . . 5 |- ( w = W -> ( x e. ( LSubSp ` w ) |-> ( x ( proj1 ` w ) ( ( ocv ` w ) ` x ) ) ) = ( x e. L |-> ( x P ( ._|_ ` x ) ) ) )
15		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
16		pjfval.v	. . . . . . . 8 |- V = ( Base ` W )
17	15, 16	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
18	17, 17	oveq12d 6668	. . . . . 6 |- ( w = W -> ( ( Base ` w ) ^m ( Base ` w ) ) = ( V ^m V ) )
19	18	xpeq2d 5139	. . . . 5 |- ( w = W -> ( _V X. ( ( Base ` w ) ^m ( Base ` w ) ) ) = ( _V X. ( V ^m V ) ) )
20	14, 19	ineq12d 3815	. . . 4 |- ( w = W -> ( ( x e. ( LSubSp ` w ) |-> ( x ( proj1 ` w ) ( ( ocv ` w ) ` x ) ) ) i^i ( _V X. ( ( Base ` w ) ^m ( Base ` w ) ) ) ) = ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) )
21		df-pj 20047	. . . 4 |- proj = ( w e. _V |-> ( ( x e. ( LSubSp ` w ) |-> ( x ( proj1 ` w ) ( ( ocv ` w ) ` x ) ) ) i^i ( _V X. ( ( Base ` w ) ^m ( Base ` w ) ) ) ) )
22		fvex 6201	. . . . . . . 8 |- ( LSubSp ` W ) e. _V
23	3, 22	eqeltri 2697	. . . . . . 7 |- L e. _V
24	23	inex1 4799	. . . . . 6 |- ( L i^i _V ) e. _V
25		ovex 6678	. . . . . . 7 |- ( V ^m V ) e. _V
26	25	inex2 4800	. . . . . 6 |- ( _V i^i ( V ^m V ) ) e. _V
27	24, 26	xpex 6962	. . . . 5 |- ( ( L i^i _V ) X. ( _V i^i ( V ^m V ) ) ) e. _V
28		eqid 2622	. . . . . . . 8 |- ( x e. L |-> ( x P ( ._|_ ` x ) ) ) = ( x e. L |-> ( x P ( ._|_ ` x ) ) )
29		ovexd 6680	. . . . . . . 8 |- ( x e. L -> ( x P ( ._|_ ` x ) ) e. _V )
30	28, 29	fmpti 6383	. . . . . . 7 |- ( x e. L |-> ( x P ( ._|_ ` x ) ) ) : L --> _V
31		fssxp 6060	. . . . . . 7 |- ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) : L --> _V -> ( x e. L |-> ( x P ( ._|_ ` x ) ) ) C_ ( L X. _V ) )
32		ssrin 3838	. . . . . . 7 |- ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) C_ ( L X. _V ) -> ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) C_ ( ( L X. _V ) i^i ( _V X. ( V ^m V ) ) ) )
33	30, 31, 32	mp2b 10	. . . . . 6 |- ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) C_ ( ( L X. _V ) i^i ( _V X. ( V ^m V ) ) )
34		inxp 5254	. . . . . 6 |- ( ( L X. _V ) i^i ( _V X. ( V ^m V ) ) ) = ( ( L i^i _V ) X. ( _V i^i ( V ^m V ) ) )
35	33, 34	sseqtri 3637	. . . . 5 |- ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) C_ ( ( L i^i _V ) X. ( _V i^i ( V ^m V ) ) )
36	27, 35	ssexi 4803	. . . 4 |- ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) e. _V
37	20, 21, 36	fvmpt 6282	. . 3 |- ( W e. _V -> ( proj ` W ) = ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) )
38		fvprc 6185	. . . 4 |- ( -. W e. _V -> ( proj ` W ) = (/) )
39		inss1 3833	. . . . 5 |- ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) C_ ( x e. L |-> ( x P ( ._|_ ` x ) ) )
40		fvprc 6185	. . . . . . . 8 |- ( -. W e. _V -> ( LSubSp ` W ) = (/) )
41	3, 40	syl5eq 2668	. . . . . . 7 |- ( -. W e. _V -> L = (/) )
42	41	mpteq1d 4738	. . . . . 6 |- ( -. W e. _V -> ( x e. L |-> ( x P ( ._|_ ` x ) ) ) = ( x e. (/) |-> ( x P ( ._|_ ` x ) ) ) )
43		mpt0 6021	. . . . . 6 |- ( x e. (/) |-> ( x P ( ._|_ ` x ) ) ) = (/)
44	42, 43	syl6eq 2672	. . . . 5 |- ( -. W e. _V -> ( x e. L |-> ( x P ( ._|_ ` x ) ) ) = (/) )
45		sseq0 3975	. . . . 5 |- ( ( ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) C_ ( x e. L |-> ( x P ( ._|_ ` x ) ) ) /\ ( x e. L |-> ( x P ( ._|_ ` x ) ) ) = (/) ) -> ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) = (/) )
46	39, 44, 45	sylancr 695	. . . 4 |- ( -. W e. _V -> ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) = (/) )
47	38, 46	eqtr4d 2659	. . 3 |- ( -. W e. _V -> ( proj ` W ) = ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) )
48	37, 47	pm2.61i 176	. 2 |- ( proj ` W ) = ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) )
49	1, 48	eqtri 2644	1 |- K = ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   proj1cpj1 18050   LSubSpclss 18932   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-pj 20047

This theorem is referenced by:  pjdm  20051  pjpm  20052  pjfval2  20053