Theorem dflinc2 48598

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 48594	. 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 8800	. . . . . . . 8 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) → 𝑠 Fn 𝑣)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣)
4		fnresi 6619	. . . . . . . 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 3442	. . . . . . . 8 ⊢ 𝑣 ∈ V
7	6	a1i 11	. . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8		inidm 4177	. . . . . . 7 ⊢ (𝑣 ∩ 𝑣) = 𝑣
9		eqidd 2735	. . . . . . 7 ⊢ (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖 ∈ 𝑣) → (𝑠‘𝑖) = (𝑠‘𝑖))
10		fvresi 7117	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖)
11	10	adantl 481	. . . . . . 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 7629	. . . . . 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 2740	. . . . 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 7372	. . . 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 7434	. . 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 5192	. 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 2757	1 ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘f ( ·𝑠 ‘𝑚)( I ↾ 𝑣)))))

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  𝒫 cpw 4552   ↦ cmpt 5177   I cid 5516   ↾ cres 5624   Fn wfn 6485  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358   ∘_f cof 7618   ↑_m cmap 8761  Basecbs 17134  Scalarcsca 17178   ·_𝑠 cvsca 17179   Σ_g cgsu 17358   linC clinc 48592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-1st 7931  df-2nd 7932  df-map 8763  df-linc 48594

This theorem is referenced by: (None)