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Theorem lincop 46390
Description: A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
lincop (𝑀 ∈ 𝑋 β†’ ( linC β€˜π‘€) = (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
Distinct variable groups:   𝑀,𝑠,𝑣,π‘₯   𝑣,𝑋
Allowed substitution hints:   𝑋(π‘₯,𝑠)

Proof of Theorem lincop
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 df-linc 46388 . 2 linC = (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)))))
2 2fveq3 6844 . . . 4 (π‘š = 𝑀 β†’ (Baseβ€˜(Scalarβ€˜π‘š)) = (Baseβ€˜(Scalarβ€˜π‘€)))
32oveq1d 7366 . . 3 (π‘š = 𝑀 β†’ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) = ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣))
4 fveq2 6839 . . . 4 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = (Baseβ€˜π‘€))
54pweqd 4575 . . 3 (π‘š = 𝑀 β†’ 𝒫 (Baseβ€˜π‘š) = 𝒫 (Baseβ€˜π‘€))
6 id 22 . . . 4 (π‘š = 𝑀 β†’ π‘š = 𝑀)
7 fveq2 6839 . . . . . 6 (π‘š = 𝑀 β†’ ( ·𝑠 β€˜π‘š) = ( ·𝑠 β€˜π‘€))
87oveqd 7368 . . . . 5 (π‘š = 𝑀 β†’ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯) = ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))
98mpteq2dv 5205 . . . 4 (π‘š = 𝑀 β†’ (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)) = (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))
106, 9oveq12d 7369 . . 3 (π‘š = 𝑀 β†’ (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯))) = (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
113, 5, 10mpoeq123dv 7426 . 2 (π‘š = 𝑀 β†’ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)))) = (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
12 elex 3461 . 2 (𝑀 ∈ 𝑋 β†’ 𝑀 ∈ V)
13 fvex 6852 . . . 4 (Baseβ€˜π‘€) ∈ V
1413pwex 5333 . . 3 𝒫 (Baseβ€˜π‘€) ∈ V
15 ovexd 7386 . . . 4 (𝑀 ∈ 𝑋 β†’ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣) ∈ V)
1615ralrimivw 3145 . . 3 (𝑀 ∈ 𝑋 β†’ βˆ€π‘£ ∈ 𝒫 (Baseβ€˜π‘€)((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣) ∈ V)
17 eqid 2737 . . . 4 (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))) = (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
1817mpoexxg2 46314 . . 3 ((𝒫 (Baseβ€˜π‘€) ∈ V ∧ βˆ€π‘£ ∈ 𝒫 (Baseβ€˜π‘€)((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣) ∈ V) β†’ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))) ∈ V)
1914, 16, 18sylancr 587 . 2 (𝑀 ∈ 𝑋 β†’ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))) ∈ V)
201, 11, 12, 19fvmptd3 6968 1 (𝑀 ∈ 𝑋 β†’ ( linC β€˜π‘€) = (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑀 Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆ€wral 3062  Vcvv 3443  π’« cpw 4558   ↦ cmpt 5186  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353   ↑m cmap 8723  Basecbs 17043  Scalarcsca 17096   ·𝑠 cvsca 17097   Ξ£g cgsu 17282   linC clinc 46386
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-rep 5240  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-reu 3352  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-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-linc 46388
This theorem is referenced by:  lincval  46391
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