Theorem scafeqg 14443

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 14441	. . 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 6161	. . 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 2268	1 |- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .xb = .x. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   X. cxp 4746   Fn wfn 5346   ` cfv 5351  (class class class)co 6049   e. cmpo 6051   Basecbs 13201  Scalarcsca 13282   .scvsca 13283   .sfcscaf 14423

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-ndx 13204  df-slot 13205  df-base 13207  df-sca 13295  df-scaf 14425

This theorem is referenced by: (None)