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Theorem lincop 41515
Description: A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
lincop (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
Distinct variable groups:   𝑀,𝑠,𝑣,𝑥   𝑣,𝑋
Allowed substitution hints:   𝑋(𝑥,𝑠)

Proof of Theorem lincop
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 df-linc 41513 . . 3 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))
21a1i 11 . 2 (𝑀𝑋 → linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥))))))
3 fveq2 6158 . . . . . 6 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
43fveq2d 6162 . . . . 5 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
54oveq1d 6630 . . . 4 (𝑚 = 𝑀 → ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣))
6 fveq2 6158 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
76pweqd 4141 . . . 4 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
8 id 22 . . . . 5 (𝑚 = 𝑀𝑚 = 𝑀)
9 fveq2 6158 . . . . . . 7 (𝑚 = 𝑀 → ( ·𝑠𝑚) = ( ·𝑠𝑀))
109oveqd 6632 . . . . . 6 (𝑚 = 𝑀 → ((𝑠𝑥)( ·𝑠𝑚)𝑥) = ((𝑠𝑥)( ·𝑠𝑀)𝑥))
1110mpteq2dv 4715 . . . . 5 (𝑚 = 𝑀 → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)) = (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))
128, 11oveq12d 6633 . . . 4 (𝑚 = 𝑀 → (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥))) = (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
135, 7, 12mpt2eq123dv 6682 . . 3 (𝑚 = 𝑀 → (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
1413adantl 482 . 2 ((𝑀𝑋𝑚 = 𝑀) → (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
15 elex 3202 . 2 (𝑀𝑋𝑀 ∈ V)
16 fvex 6168 . . . 4 (Base‘𝑀) ∈ V
1716pwex 4818 . . 3 𝒫 (Base‘𝑀) ∈ V
18 ovexd 6645 . . . 4 (𝑀𝑋 → ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣) ∈ V)
1918ralrimivw 2963 . . 3 (𝑀𝑋 → ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣) ∈ V)
20 eqid 2621 . . . 4 (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
2120mpt2exxg2 41434 . . 3 ((𝒫 (Base‘𝑀) ∈ V ∧ ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣) ∈ V) → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) ∈ V)
2217, 19, 21sylancr 694 . 2 (𝑀𝑋 → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) ∈ V)
232, 14, 15, 22fvmptd 6255 1 (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  𝒫 cpw 4136  cmpt 4683  cfv 5857  (class class class)co 6615  cmpt2 6617  𝑚 cmap 7817  Basecbs 15800  Scalarcsca 15884   ·𝑠 cvsca 15885   Σg cgsu 16041   linC clinc 41511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-linc 41513
This theorem is referenced by:  lincval  41516
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