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Theorem scafeq 20902
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 scaffval.b . . 3 𝐵 = (Base‘𝑊)
2 scaffval.f . . 3 𝐹 = (Scalar‘𝑊)
3 scaffval.k . . 3 𝐾 = (Base‘𝐹)
4 scaffval.a . . 3 = ( ·sf𝑊)
5 scaffval.s . . 3 · = ( ·𝑠𝑊)
61, 2, 3, 4, 5scaffval 20900 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
7 fnov 7581 . . 3 ( · Fn (𝐾 × 𝐵) ↔ · = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
87biimpi 216 . 2 ( · Fn (𝐾 × 𝐵) → · = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
96, 8eqtr4id 2799 1 ( · Fn (𝐾 × 𝐵) → = · )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537   × cxp 5698   Fn wfn 6568  cfv 6573  (class class class)co 7448  cmpo 7450  Basecbs 17258  Scalarcsca 17314   ·𝑠 cvsca 17315   ·sf cscaf 20881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-scaf 20883
This theorem is referenced by: (None)
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