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Theorem dflinc2 47348
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 47344 . 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 8858 . . . . . . . 8 (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) β†’ 𝑠 Fn 𝑣)
32adantr 480 . . . . . . 7 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ 𝑠 Fn 𝑣)
4 fnresi 6672 . . . . . . . 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 3472 . . . . . . . 8 𝑣 ∈ V
76a1i 11 . . . . . . 7 ((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ 𝑣 ∈ V)
8 inidm 4213 . . . . . . 7 (𝑣 ∩ 𝑣) = 𝑣
9 eqidd 2727 . . . . . . 7 (((𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Baseβ€˜π‘š)) ∧ 𝑖 ∈ 𝑣) β†’ (π‘ β€˜π‘–) = (π‘ β€˜π‘–))
10 fvresi 7166 . . . . . . . 8 (𝑖 ∈ 𝑣 β†’ (( I β†Ύ 𝑣)β€˜π‘–) = 𝑖)
1110adantl 481 . . . . . . 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 7675 . . . . . 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 2732 . . . . 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 7420 . . . 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 7482 . . 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 5246 . 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 2754 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  π’« cpw 4597   ↦ cmpt 5224   I cid 5566   β†Ύ cres 5671   Fn wfn 6531  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406   ∘f cof 7664   ↑m cmap 8819  Basecbs 17150  Scalarcsca 17206   ·𝑠 cvsca 17207   Ξ£g cgsu 17392   linC clinc 47342
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666  df-1st 7971  df-2nd 7972  df-map 8821  df-linc 47344
This theorem is referenced by: (None)
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