Theorem dflinc2 48386

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 48382	. 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 8879	. . . . . . . 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 6667	. . . . . . . 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 3463	. . . . . . . 8 ⊢ 𝑣 ∈ V
7	6	a1i 11	. . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8		inidm 4202	. . . . . . 7 ⊢ (𝑣 ∩ 𝑣) = 𝑣
9		eqidd 2736	. . . . . . 7 ⊢ (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖 ∈ 𝑣) → (𝑠‘𝑖) = (𝑠‘𝑖))
10		fvresi 7165	. . . . . . . 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 7680	. . . . . 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 2741	. . . . 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 7421	. . . 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 7485	. . 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 5217	. 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 2758	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 1540   ∈ wcel 2108  Vcvv 3459  𝒫 cpw 4575   ↦ cmpt 5201   I cid 5547   ↾ cres 5656   Fn wfn 6526  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407   ∘_f cof 7669   ↑_m cmap 8840  Basecbs 17228  Scalarcsca 17274   ·_𝑠 cvsca 17275   Σ_g cgsu 17454   linC clinc 48380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-1st 7988  df-2nd 7989  df-map 8842  df-linc 48382

This theorem is referenced by: (None)