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Theorem scaffval 20293
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 6839 . . . . . . . 8 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
3 scaffval.f . . . . . . . 8 𝐹 = (Scalarβ€˜π‘Š)
42, 3eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝐹)
54fveq2d 6843 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜πΉ))
6 scaffval.k . . . . . 6 𝐾 = (Baseβ€˜πΉ)
75, 6eqtr4di 2795 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = 𝐾)
8 fveq2 6839 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
9 scaffval.b . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
108, 9eqtr4di 2795 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝐡)
11 fveq2 6839 . . . . . . 7 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘Š))
12 scaffval.s . . . . . . 7 Β· = ( ·𝑠 β€˜π‘Š)
1311, 12eqtr4di 2795 . . . . . 6 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = Β· )
1413oveqd 7368 . . . . 5 (𝑀 = π‘Š β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) = (π‘₯ Β· 𝑦))
157, 10, 14mpoeq123dv 7426 . . . 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 20278 . . . 4 Β·sf = (𝑀 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)), 𝑦 ∈ (Baseβ€˜π‘€) ↦ (π‘₯( ·𝑠 β€˜π‘€)𝑦)))
176fvexi 6853 . . . . 5 𝐾 ∈ V
189fvexi 6853 . . . . 5 𝐡 ∈ V
1912fvexi 6853 . . . . . . 7 Β· ∈ V
2019rnex 7841 . . . . . 6 ran Β· ∈ V
21 p0ex 5337 . . . . . 6 {βˆ…} ∈ V
2220, 21unex 7672 . . . . 5 (ran Β· βˆͺ {βˆ…}) ∈ V
23 df-ov 7354 . . . . . . 7 (π‘₯ Β· 𝑦) = ( Β· β€˜βŸ¨π‘₯, π‘¦βŸ©)
24 fvrn0 6869 . . . . . . 7 ( Β· β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ (ran Β· βˆͺ {βˆ…})
2523, 24eqeltri 2834 . . . . . 6 (π‘₯ Β· 𝑦) ∈ (ran Β· βˆͺ {βˆ…})
2625rgen2w 3067 . . . . 5 βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) ∈ (ran Β· βˆͺ {βˆ…})
2717, 18, 22, 26mpoexw 8003 . . . 4 (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)) ∈ V
2815, 16, 27fvmpt 6945 . . 3 (π‘Š ∈ V β†’ ( Β·sf β€˜π‘Š) = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)))
29 fvprc 6831 . . . 4 (Β¬ π‘Š ∈ V β†’ ( Β·sf β€˜π‘Š) = βˆ…)
30 fvprc 6831 . . . . . . 7 (Β¬ π‘Š ∈ V β†’ (Baseβ€˜π‘Š) = βˆ…)
319, 30eqtrid 2789 . . . . . 6 (Β¬ π‘Š ∈ V β†’ 𝐡 = βˆ…)
3231olcd 872 . . . . 5 (Β¬ π‘Š ∈ V β†’ (𝐾 = βˆ… ∨ 𝐡 = βˆ…))
33 0mpo0 7434 . . . . 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 2780 . . 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 2765 1 βˆ™ = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3443   βˆͺ cun 3906  βˆ…c0 4280  {csn 4584  βŸ¨cop 4590  ran crn 5632  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  Basecbs 17043  Scalarcsca 17096   ·𝑠 cvsca 17097   Β·sf cscaf 20276
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-scaf 20278
This theorem is referenced by:  scafval  20294  scafeq  20295  scaffn  20296  lmodscaf  20297  rlmscaf  20631
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