Theorem scafeqg 13804

Description: If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (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
scafeqg	|- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .xb = .x. )

Proof of Theorem scafeqg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.b	. . . 4 |- B = ( Base ` W )
2		scaffval.f	. . . 4 |- F = (Scalar ` W )
3		scaffval.k	. . . 4 |- K = ( Base ` F )
4		scaffval.a	. . . 4 |- .xb = ( .sf ` W )
5		scaffval.s	. . . 4 |- .x. = ( .s ` W )
6	1, 2, 3, 4, 5	scaffvalg 13802	. . 3 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
7	6	adantr 276	. 2 |- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
8		fnovim 6027	. . 3 |- ( .x. Fn ( K X. B ) -> .x. = ( x e. K , y e. B |-> ( x .x. y ) ) )
9	8	adantl 277	. 2 |- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .x. = ( x e. K , y e. B |-> ( x .x. y ) ) )
10	7, 9	eqtr4d 2229	1 |- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .xb = .x. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   X. cxp 4657   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   Basecbs 12618  Scalarcsca 12698   .scvsca 12699   .sfcscaf 13784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-ndx 12621  df-slot 12622  df-base 12624  df-sca 12711  df-scaf 13786

This theorem is referenced by: (None)