Theorem dflinc2 49247

Description: Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)

Assertion

Ref	Expression
dflinc2	⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘f ( ·𝑠 ‘𝑚)( I ↾ 𝑣)))))

Distinct variable group:   𝑚,𝑠,𝑣

Proof of Theorem dflinc2

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-linc 49243	. 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 8868	. . . . . . . 8 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) → 𝑠 Fn 𝑣)
3	2	adantr 486	. . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣)
4		fnresi 6668	. . . . . . . 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 3461	. . . . . . . 8 ⊢ 𝑣 ∈ V
7	6	a1i 11	. . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8		inidm 4179	. . . . . . 7 ⊢ (𝑣 ∩ 𝑣) = 𝑣
9		eqidd 2766	. . . . . . 7 ⊢ (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖 ∈ 𝑣) → (𝑠‘𝑖) = (𝑠‘𝑖))
10		fvresi 7177	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖)
11	10	adantl 487	. . . . . . 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 7693	. . . . . 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 2771	. . . . 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 7435	. . . 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	mpoeq3ia 7497	. . 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 5209	. 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 2788	1 ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘f ( ·𝑠 ‘𝑚)( I ↾ 𝑣)))))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  𝒫 cpw 4564   ↦ cmpt 5194   I cid 5557   ↾ cres 5665   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421   ∘_f cof 7682   ↑_m cmap 8830  Basecbs 17291  Scalarcsca 17335   ·_𝑠 cvsca 17336   Σ_g cgsu 17515   linC clinc 49241

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-1st 7992  df-2nd 7993  df-map 8832  df-linc 49243

This theorem is used by: (None)