Theorem scafeq 18883

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
scafeq	|- ( .x. Fn ( K X. B ) -> .xb = .x. )

Proof of Theorem scafeq

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

Step	Hyp	Ref	Expression
1		fnov 6768	. . 3 |- ( .x. Fn ( K X. B ) <-> .x. = ( x e. K , y e. B |-> ( x .x. y ) ) )
2	1	biimpi 206	. 2 |- ( .x. Fn ( K X. B ) -> .x. = ( x e. K , y e. B |-> ( x .x. y ) ) )
3		scaffval.b	. . 3 |- B = ( Base ` W )
4		scaffval.f	. . 3 |- F = (Scalar ` W )
5		scaffval.k	. . 3 |- K = ( Base ` F )
6		scaffval.a	. . 3 |- .xb = ( .sf ` W )
7		scaffval.s	. . 3 |- .x. = ( .s ` W )
8	3, 4, 5, 6, 7	scaffval 18881	. 2 |- .xb = ( x e. K , y e. B |-> ( x .x. y ) )
9	2, 8	syl6reqr 2675	1 |- ( .x. Fn ( K X. B ) -> .xb = .x. )

Syntax hints:   -> wi 4   = wceq 1483   X. cxp 5112   Fn wfn 5883   ` 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: (None)