Theorem scafeqg 14325

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 14323	. . 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 6130	. . 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 2267	1 |- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .xb = .x. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   X. cxp 4723   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   e. cmpo 6020   Basecbs 13084  Scalarcsca 13165   .scvsca 13166   .sfcscaf 14305

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-ndx 13087  df-slot 13088  df-base 13090  df-sca 13178  df-scaf 14307

This theorem is referenced by: (None)