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Mirrors > Home > MPE Home > Th. List > Mathboxes > dflinc2 | Structured version Visualization version GIF version |
Description: Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.) |
Ref | Expression |
---|---|
dflinc2 | ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-linc 43035 | . 2 ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖 ∈ 𝑣 ↦ ((𝑠‘𝑖)( ·𝑠 ‘𝑚)𝑖))))) | |
2 | elmapfn 8145 | . . . . . . . 8 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) → 𝑠 Fn 𝑣) | |
3 | 2 | adantr 474 | . . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣) |
4 | fnresi 6241 | . . . . . . . 8 ⊢ ( I ↾ 𝑣) Fn 𝑣 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → ( I ↾ 𝑣) Fn 𝑣) |
6 | vex 3417 | . . . . . . . 8 ⊢ 𝑣 ∈ V | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V) |
8 | inidm 4047 | . . . . . . 7 ⊢ (𝑣 ∩ 𝑣) = 𝑣 | |
9 | eqidd 2826 | . . . . . . 7 ⊢ (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖 ∈ 𝑣) → (𝑠‘𝑖) = (𝑠‘𝑖)) | |
10 | fvresi 6691 | . . . . . . . 8 ⊢ (𝑖 ∈ 𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖) | |
11 | 10 | adantl 475 | . . . . . . 7 ⊢ (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖 ∈ 𝑣) → (( I ↾ 𝑣)‘𝑖) = 𝑖) |
12 | 3, 5, 7, 7, 8, 9, 11 | offval 7164 | . . . . . 6 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣)) = (𝑖 ∈ 𝑣 ↦ ((𝑠‘𝑖)( ·𝑠 ‘𝑚)𝑖))) |
13 | 12 | eqcomd 2831 | . . . . 5 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑖 ∈ 𝑣 ↦ ((𝑠‘𝑖)( ·𝑠 ‘𝑚)𝑖)) = (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣))) |
14 | 13 | oveq2d 6921 | . . . 4 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑚 Σg (𝑖 ∈ 𝑣 ↦ ((𝑠‘𝑖)( ·𝑠 ‘𝑚)𝑖))) = (𝑚 Σg (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣)))) |
15 | 14 | mpt2eq3ia 6980 | . . 3 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖 ∈ 𝑣 ↦ ((𝑠‘𝑖)( ·𝑠 ‘𝑚)𝑖)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣)))) |
16 | 15 | mpteq2i 4964 | . 2 ⊢ (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖 ∈ 𝑣 ↦ ((𝑠‘𝑖)( ·𝑠 ‘𝑚)𝑖))))) = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣))))) |
17 | 1, 16 | eqtri 2849 | 1 ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣))))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 𝒫 cpw 4378 ↦ cmpt 4952 I cid 5249 ↾ cres 5344 Fn wfn 6118 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 ∘𝑓 cof 7155 ↑𝑚 cmap 8122 Basecbs 16222 Scalarcsca 16308 ·𝑠 cvsca 16309 Σg cgsu 16454 linC clinc 43033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-1st 7428 df-2nd 7429 df-map 8124 df-linc 43035 |
This theorem is referenced by: (None) |
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