Theorem ipffval 19993

Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)

Hypotheses

Ref	Expression
ipffval.1	|- V = ( Base ` W )
ipffval.2	|- ., = ( .i ` W )
ipffval.3	|- .x. = ( .if ` W )

Assertion

Ref	Expression
ipffval	|- .x. = ( x e. V , y e. V |-> ( x ., y ) )

Distinct variable groups:   x, y, .,   x, V, y   x, W, y

Allowed substitution hints:   .x. ( x, y)

Proof of Theorem ipffval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ipffval.3	. 2 |- .x. = ( .if ` W )
2		fveq2 6191	. . . . . 6 |- ( g = W -> ( Base ` g ) = ( Base ` W ) )
3		ipffval.1	. . . . . 6 |- V = ( Base ` W )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( g = W -> ( Base ` g ) = V )
5		fveq2 6191	. . . . . . 7 |- ( g = W -> ( .i ` g ) = ( .i ` W ) )
6		ipffval.2	. . . . . . 7 |- ., = ( .i ` W )
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( g = W -> ( .i ` g ) = ., )
8	7	oveqd 6667	. . . . 5 |- ( g = W -> ( x ( .i ` g ) y ) = ( x ., y ) )
9	4, 4, 8	mpt2eq123dv 6717	. . . 4 |- ( g = W -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( .i ` g ) y ) ) = ( x e. V , y e. V |-> ( x ., y ) ) )
10		df-ipf 19972	. . . 4 |- .if = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( .i ` g ) y ) ) )
11		df-ov 6653	. . . . . . . 8 |- ( x ., y ) = ( ., ` <. x , y >. )
12		fvrn0 6216	. . . . . . . 8 |- ( ., ` <. x , y >. ) e. ( ran ., u. { (/) } )
13	11, 12	eqeltri 2697	. . . . . . 7 |- ( x ., y ) e. ( ran ., u. { (/) } )
14	13	rgen2w 2925	. . . . . 6 |- A. x e. V A. y e. V ( x ., y ) e. ( ran ., u. { (/) } )
15		eqid 2622	. . . . . . 7 |- ( x e. V , y e. V |-> ( x ., y ) ) = ( x e. V , y e. V |-> ( x ., y ) )
16	15	fmpt2 7237	. . . . . 6 |- ( A. x e. V A. y e. V ( x ., y ) e. ( ran ., u. { (/) } ) <-> ( x e. V , y e. V |-> ( x ., y ) ) : ( V X. V ) --> ( ran ., u. { (/) } ) )
17	14, 16	mpbi 220	. . . . 5 |- ( x e. V , y e. V |-> ( x ., y ) ) : ( V X. V ) --> ( ran ., u. { (/) } )
18		fvex 6201	. . . . . . 7 |- ( Base ` W ) e. _V
19	3, 18	eqeltri 2697	. . . . . 6 |- V e. _V
20	19, 19	xpex 6962	. . . . 5 |- ( V X. V ) e. _V
21		fvex 6201	. . . . . . . 8 |- ( .i ` W ) e. _V
22	6, 21	eqeltri 2697	. . . . . . 7 |- ., e. _V
23	22	rnex 7100	. . . . . 6 |- ran ., e. _V
24		p0ex 4853	. . . . . 6 |- { (/) } e. _V
25	23, 24	unex 6956	. . . . 5 |- ( ran ., u. { (/) } ) e. _V
26		fex2 7121	. . . . 5 |- ( ( ( x e. V , y e. V |-> ( x ., y ) ) : ( V X. V ) --> ( ran ., u. { (/) } ) /\ ( V X. V ) e. _V /\ ( ran ., u. { (/) } ) e. _V ) -> ( x e. V , y e. V |-> ( x ., y ) ) e. _V )
27	17, 20, 25, 26	mp3an 1424	. . . 4 |- ( x e. V , y e. V |-> ( x ., y ) ) e. _V
28	9, 10, 27	fvmpt 6282	. . 3 |- ( W e. _V -> ( .if ` W ) = ( x e. V , y e. V |-> ( x ., y ) ) )
29		fvprc 6185	. . . . 5 |- ( -. W e. _V -> ( .if ` W ) = (/) )
30		mpt20 6725	. . . . 5 |- ( x e. (/) , y e. (/) |-> ( x ., y ) ) = (/)
31	29, 30	syl6eqr 2674	. . . 4 |- ( -. W e. _V -> ( .if ` W ) = ( x e. (/) , y e. (/) |-> ( x ., y ) ) )
32		fvprc 6185	. . . . . 6 |- ( -. W e. _V -> ( Base ` W ) = (/) )
33	3, 32	syl5eq 2668	. . . . 5 |- ( -. W e. _V -> V = (/) )
34		mpt2eq12 6715	. . . . 5 |- ( ( V = (/) /\ V = (/) ) -> ( x e. V , y e. V |-> ( x ., y ) ) = ( x e. (/) , y e. (/) |-> ( x ., y ) ) )
35	33, 33, 34	syl2anc 693	. . . 4 |- ( -. W e. _V -> ( x e. V , y e. V |-> ( x ., y ) ) = ( x e. (/) , y e. (/) |-> ( x ., y ) ) )
36	31, 35	eqtr4d 2659	. . 3 |- ( -. W e. _V -> ( .if ` W ) = ( x e. V , y e. V |-> ( x ., y ) ) )
37	28, 36	pm2.61i 176	. 2 |- ( .if ` W ) = ( x e. V , y e. V |-> ( x ., y ) )
38	1, 37	eqtri 2644	1 |- .x. = ( x e. V , y e. V |-> ( x ., y ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   <.cop 4183   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   .icip 15946   .ifcipf 19970

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-csb 3534  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-iun 4522  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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ipf 19972

This theorem is referenced by:  ipfval  19994  ipfeq  19995  ipffn  19996  phlipf  19997  phssip  20003