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Theorem dflinc2 47556
Description: Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
Assertion
Ref Expression
dflinc2 linC = (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (𝑠 ∘f ( ·𝑠 β€˜π‘š)( I β†Ύ 𝑣)))))
Distinct variable group:   π‘š,𝑠,𝑣

Proof of Theorem dflinc2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 df-linc 47552 . 2 linC = (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (𝑖 ∈ 𝑣 ↦ ((π‘ β€˜π‘–)( ·𝑠 β€˜π‘š)𝑖)))))
2 elmapfn 8890 . . . . . . . 8 (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) β†’ 𝑠 Fn 𝑣)
32adantr 479 . . . . . . 7 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ 𝑠 Fn 𝑣)
4 fnresi 6689 . . . . . . . 8 ( I β†Ύ 𝑣) Fn 𝑣
54a1i 11 . . . . . . 7 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ ( I β†Ύ 𝑣) Fn 𝑣)
6 vex 3477 . . . . . . . 8 𝑣 ∈ V
76a1i 11 . . . . . . 7 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ 𝑣 ∈ V)
8 inidm 4221 . . . . . . 7 (𝑣 ∩ 𝑣) = 𝑣
9 eqidd 2729 . . . . . . 7 (((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) ∧ 𝑖 ∈ 𝑣) β†’ (π‘ β€˜π‘–) = (π‘ β€˜π‘–))
10 fvresi 7188 . . . . . . . 8 (𝑖 ∈ 𝑣 β†’ (( I β†Ύ 𝑣)β€˜π‘–) = 𝑖)
1110adantl 480 . . . . . . 7 (((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) ∧ 𝑖 ∈ 𝑣) β†’ (( I β†Ύ 𝑣)β€˜π‘–) = 𝑖)
123, 5, 7, 7, 8, 9, 11offval 7700 . . . . . 6 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ (𝑠 ∘f ( ·𝑠 β€˜π‘š)( I β†Ύ 𝑣)) = (𝑖 ∈ 𝑣 ↦ ((π‘ β€˜π‘–)( ·𝑠 β€˜π‘š)𝑖)))
1312eqcomd 2734 . . . . 5 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ (𝑖 ∈ 𝑣 ↦ ((π‘ β€˜π‘–)( ·𝑠 β€˜π‘š)𝑖)) = (𝑠 ∘f ( ·𝑠 β€˜π‘š)( I β†Ύ 𝑣)))
1413oveq2d 7442 . . . 4 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ (π‘š Ξ£g (𝑖 ∈ 𝑣 ↦ ((π‘ β€˜π‘–)( ·𝑠 β€˜π‘š)𝑖))) = (π‘š Ξ£g (𝑠 ∘f ( ·𝑠 β€˜π‘š)( I β†Ύ 𝑣))))
1514mpoeq3ia 7504 . . 3 (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (𝑖 ∈ 𝑣 ↦ ((π‘ β€˜π‘–)( ·𝑠 β€˜π‘š)𝑖)))) = (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (𝑠 ∘f ( ·𝑠 β€˜π‘š)( I β†Ύ 𝑣))))
1615mpteq2i 5257 . 2 (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (𝑖 ∈ 𝑣 ↦ ((π‘ β€˜π‘–)( ·𝑠 β€˜π‘š)𝑖))))) = (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (𝑠 ∘f ( ·𝑠 β€˜π‘š)( I β†Ύ 𝑣)))))
171, 16eqtri 2756 1 linC = (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (𝑠 ∘f ( ·𝑠 β€˜π‘š)( I β†Ύ 𝑣)))))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473  π’« cpw 4606   ↦ cmpt 5235   I cid 5579   β†Ύ cres 5684   Fn wfn 6548  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428   ∘f cof 7689   ↑m cmap 8851  Basecbs 17187  Scalarcsca 17243   ·𝑠 cvsca 17244   Ξ£g cgsu 17429   linC clinc 47550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-1st 7999  df-2nd 8000  df-map 8853  df-linc 47552
This theorem is referenced by: (None)
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