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Theorem scafeq 19640
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 = ( ·sf𝑊)
scaffval.s · = ( ·𝑠𝑊)
Assertion
Ref Expression
scafeq ( · Fn (𝐾 × 𝐵) → = · )

Proof of Theorem scafeq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnov 7264 . . 3 ( · Fn (𝐾 × 𝐵) ↔ · = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
21biimpi 219 . 2 ( · Fn (𝐾 × 𝐵) → · = (𝑥𝐾, 𝑦𝐵 ↦ (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 = ( ·sf𝑊)
7 scaffval.s . . 3 · = ( ·𝑠𝑊)
83, 4, 5, 6, 7scaffval 19638 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
92, 8syl6reqr 2878 1 ( · Fn (𝐾 × 𝐵) → = · )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538   × cxp 5534   Fn wfn 6331  cfv 6336  (class class class)co 7138  cmpo 7140  Basecbs 16472  Scalarcsca 16557   ·𝑠 cvsca 16558   ·sf cscaf 19621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-scaf 19623
This theorem is referenced by: (None)
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