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Theorem scaffval 20634
Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
Hypotheses
Ref Expression
scaffval.b 𝐡 = (Baseβ€˜π‘Š)
scaffval.f 𝐹 = (Scalarβ€˜π‘Š)
scaffval.k 𝐾 = (Baseβ€˜πΉ)
scaffval.a βˆ™ = ( Β·sf β€˜π‘Š)
scaffval.s Β· = ( ·𝑠 β€˜π‘Š)
Assertion
Ref Expression
scaffval βˆ™ = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐾,𝑦   π‘₯, Β· ,𝑦   π‘₯,π‘Š,𝑦
Allowed substitution hints:   βˆ™ (π‘₯,𝑦)   𝐹(π‘₯,𝑦)

Proof of Theorem scaffval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 scaffval.a . 2 βˆ™ = ( Β·sf β€˜π‘Š)
2 fveq2 6890 . . . . . . . 8 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
3 scaffval.f . . . . . . . 8 𝐹 = (Scalarβ€˜π‘Š)
42, 3eqtr4di 2788 . . . . . . 7 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝐹)
54fveq2d 6894 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜πΉ))
6 scaffval.k . . . . . 6 𝐾 = (Baseβ€˜πΉ)
75, 6eqtr4di 2788 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = 𝐾)
8 fveq2 6890 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
9 scaffval.b . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
108, 9eqtr4di 2788 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝐡)
11 fveq2 6890 . . . . . . 7 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘Š))
12 scaffval.s . . . . . . 7 Β· = ( ·𝑠 β€˜π‘Š)
1311, 12eqtr4di 2788 . . . . . 6 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = Β· )
1413oveqd 7428 . . . . 5 (𝑀 = π‘Š β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) = (π‘₯ Β· 𝑦))
157, 10, 14mpoeq123dv 7486 . . . 4 (𝑀 = π‘Š β†’ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)), 𝑦 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)))
16 df-scaf 20617 . . . 4 Β·sf = (𝑀 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)), 𝑦 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)𝑦)))
176fvexi 6904 . . . . 5 𝐾 ∈ V
189fvexi 6904 . . . . 5 𝐡 ∈ V
1912fvexi 6904 . . . . . . 7 Β· ∈ V
2019rnex 7905 . . . . . 6 ran Β· ∈ V
21 p0ex 5381 . . . . . 6 {βˆ…} ∈ V
2220, 21unex 7735 . . . . 5 (ran Β· βˆͺ {βˆ…}) ∈ V
23 df-ov 7414 . . . . . . 7 (π‘₯ Β· 𝑦) = ( Β· β€˜βŸ¨π‘₯, π‘¦βŸ©)
24 fvrn0 6920 . . . . . . 7 ( Β· β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ (ran Β· βˆͺ {βˆ…})
2523, 24eqeltri 2827 . . . . . 6 (π‘₯ Β· 𝑦) ∈ (ran Β· βˆͺ {βˆ…})
2625rgen2w 3064 . . . . 5 βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) ∈ (ran Β· βˆͺ {βˆ…})
2717, 18, 22, 26mpoexw 8067 . . . 4 (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)) ∈ V
2815, 16, 27fvmpt 6997 . . 3 (π‘Š ∈ V β†’ ( Β·sf β€˜π‘Š) = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)))
29 fvprc 6882 . . . 4 (Β¬ π‘Š ∈ V β†’ ( Β·sf β€˜π‘Š) = βˆ…)
30 fvprc 6882 . . . . . . 7 (Β¬ π‘Š ∈ V β†’ (Baseβ€˜π‘Š) = βˆ…)
319, 30eqtrid 2782 . . . . . 6 (Β¬ π‘Š ∈ V β†’ 𝐡 = βˆ…)
3231olcd 870 . . . . 5 (Β¬ π‘Š ∈ V β†’ (𝐾 = βˆ… ∨ 𝐡 = βˆ…))
33 0mpo0 7494 . . . . 5 ((𝐾 = βˆ… ∨ 𝐡 = βˆ…) β†’ (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)) = βˆ…)
3432, 33syl 17 . . . 4 (Β¬ π‘Š ∈ V β†’ (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)) = βˆ…)
3529, 34eqtr4d 2773 . . 3 (Β¬ π‘Š ∈ V β†’ ( Β·sf β€˜π‘Š) = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)))
3628, 35pm2.61i 182 . 2 ( Β·sf β€˜π‘Š) = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦))
371, 36eqtri 2758 1 βˆ™ = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∨ wo 843   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βˆͺ cun 3945  βˆ…c0 4321  {csn 4627  βŸ¨cop 4633  ran crn 5676  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17148  Scalarcsca 17204   ·𝑠 cvsca 17205   Β·sf cscaf 20615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-scaf 20617
This theorem is referenced by:  scafval  20635  scafeq  20636  scaffn  20637  lmodscaf  20638  rlmscaf  20976
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