Theorem scaffval 18881

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
scaffval.b	|- B = ( Base ` W )
scaffval.f	|- F = (Scalar ` W )
scaffval.k	|- K = ( Base ` F )
scaffval.a	|- .xb = ( .sf ` W )
scaffval.s	|- .x. = ( .s ` W )

Assertion

Ref	Expression
scaffval	|- .xb = ( x e. K , y e. B |-> ( x .x. y ) )

Distinct variable groups:   x, y, B   x, K, y   x, .x. , y   x, W, y

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

Proof of Theorem scaffval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.a	. 2 |- .xb = ( .sf ` W )
2		fveq2 6191	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
3		scaffval.f	. . . . . . . 8 |- F = (Scalar ` W )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
5	4	fveq2d 6195	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
6		scaffval.k	. . . . . 6 |- K = ( Base ` F )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
8		fveq2 6191	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
9		scaffval.b	. . . . . 6 |- B = ( Base ` W )
10	8, 9	syl6eqr 2674	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
11		fveq2 6191	. . . . . . 7 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
12		scaffval.s	. . . . . . 7 |- .x. = ( .s ` W )
13	11, 12	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( .s ` w ) = .x. )
14	13	oveqd 6667	. . . . 5 |- ( w = W -> ( x ( .s ` w ) y ) = ( x .x. y ) )
15	7, 10, 14	mpt2eq123dv 6717	. . . 4 |- ( w = W -> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) = ( x e. K , y e. B |-> ( x .x. y ) ) )
16		df-scaf 18866	. . . 4 |- .sf = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) , y e. ( Base ` w ) |-> ( x ( .s ` w ) y ) ) )
17		df-ov 6653	. . . . . . . 8 |- ( x .x. y ) = ( .x. ` <. x , y >. )
18		fvrn0 6216	. . . . . . . 8 |- ( .x. ` <. x , y >. ) e. ( ran .x. u. { (/) } )
19	17, 18	eqeltri 2697	. . . . . . 7 |- ( x .x. y ) e. ( ran .x. u. { (/) } )
20	19	rgen2w 2925	. . . . . 6 |- A. x e. K A. y e. B ( x .x. y ) e. ( ran .x. u. { (/) } )
21		eqid 2622	. . . . . . 7 |- ( x e. K , y e. B |-> ( x .x. y ) ) = ( x e. K , y e. B |-> ( x .x. y ) )
22	21	fmpt2 7237	. . . . . 6 |- ( A. x e. K A. y e. B ( x .x. y ) e. ( ran .x. u. { (/) } ) <-> ( x e. K , y e. B |-> ( x .x. y ) ) : ( K X. B ) --> ( ran .x. u. { (/) } ) )
23	20, 22	mpbi 220	. . . . 5 |- ( x e. K , y e. B |-> ( x .x. y ) ) : ( K X. B ) --> ( ran .x. u. { (/) } )
24		fvex 6201	. . . . . . 7 |- ( Base ` F ) e. _V
25	6, 24	eqeltri 2697	. . . . . 6 |- K e. _V
26		fvex 6201	. . . . . . 7 |- ( Base ` W ) e. _V
27	9, 26	eqeltri 2697	. . . . . 6 |- B e. _V
28	25, 27	xpex 6962	. . . . 5 |- ( K X. B ) e. _V
29		fvex 6201	. . . . . . . 8 |- ( .s ` W ) e. _V
30	12, 29	eqeltri 2697	. . . . . . 7 |- .x. e. _V
31	30	rnex 7100	. . . . . 6 |- ran .x. e. _V
32		p0ex 4853	. . . . . 6 |- { (/) } e. _V
33	31, 32	unex 6956	. . . . 5 |- ( ran .x. u. { (/) } ) e. _V
34		fex2 7121	. . . . 5 |- ( ( ( x e. K , y e. B |-> ( x .x. y ) ) : ( K X. B ) --> ( ran .x. u. { (/) } ) /\ ( K X. B ) e. _V /\ ( ran .x. u. { (/) } ) e. _V ) -> ( x e. K , y e. B |-> ( x .x. y ) ) e. _V )
35	23, 28, 33, 34	mp3an 1424	. . . 4 |- ( x e. K , y e. B |-> ( x .x. y ) ) e. _V
36	15, 16, 35	fvmpt 6282	. . 3 |- ( W e. _V -> ( .sf ` W ) = ( x e. K , y e. B |-> ( x .x. y ) ) )
37		fvprc 6185	. . . . 5 |- ( -. W e. _V -> ( .sf ` W ) = (/) )
38		mpt20 6725	. . . . 5 |- ( x e. (/) , y e. B |-> ( x .x. y ) ) = (/)
39	37, 38	syl6eqr 2674	. . . 4 |- ( -. W e. _V -> ( .sf ` W ) = ( x e. (/) , y e. B |-> ( x .x. y ) ) )
40		fvprc 6185	. . . . . . . . 9 |- ( -. W e. _V -> (Scalar ` W ) = (/) )
41	3, 40	syl5eq 2668	. . . . . . . 8 |- ( -. W e. _V -> F = (/) )
42	41	fveq2d 6195	. . . . . . 7 |- ( -. W e. _V -> ( Base ` F ) = ( Base ` (/) ) )
43	6, 42	syl5eq 2668	. . . . . 6 |- ( -. W e. _V -> K = ( Base ` (/) ) )
44		base0 15912	. . . . . 6 |- (/) = ( Base ` (/) )
45	43, 44	syl6eqr 2674	. . . . 5 |- ( -. W e. _V -> K = (/) )
46		eqid 2622	. . . . 5 |- B = B
47		mpt2eq12 6715	. . . . 5 |- ( ( K = (/) /\ B = B ) -> ( x e. K , y e. B |-> ( x .x. y ) ) = ( x e. (/) , y e. B |-> ( x .x. y ) ) )
48	45, 46, 47	sylancl 694	. . . 4 |- ( -. W e. _V -> ( x e. K , y e. B |-> ( x .x. y ) ) = ( x e. (/) , y e. B |-> ( x .x. y ) ) )
49	39, 48	eqtr4d 2659	. . 3 |- ( -. W e. _V -> ( .sf ` W ) = ( x e. K , y e. B |-> ( x .x. y ) ) )
50	36, 49	pm2.61i 176	. 2 |- ( .sf ` W ) = ( x e. K , y e. B |-> ( x .x. y ) )
51	1, 50	eqtri 2644	1 |- .xb = ( x e. K , y e. B |-> ( x .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  Scalarcsca 15944   .scvsca 15945   .sfcscaf 18864

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-slot 15861  df-base 15863  df-scaf 18866

This theorem is referenced by:  scafval  18882  scafeq  18883  scaffn  18884  lmodscaf  18885  rlmscaf  19208