Theorem scafeqg 14473

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 14471	. . 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 6164	. . 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 2270	1 |- ( ( W e. V /\ .x. Fn ( K X. B ) ) -> .xb = .x. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   X. cxp 4749   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   e. cmpo 6054   Basecbs 13229  Scalarcsca 13310   .scvsca 13311   .sfcscaf 14453

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-ndx 13232  df-slot 13233  df-base 13235  df-sca 13323  df-scaf 14455

This theorem is referenced by: (None)