Theorem dflinc2 42727

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 42723	. 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 8048	. . . . . . . 8 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) → 𝑠 Fn 𝑣)
3	2	adantr 472	. . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣)
4		fnresi 6169	. . . . . . . 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 3343	. . . . . . . 8 ⊢ 𝑣 ∈ V
7	6	a1i 11	. . . . . . 7 ⊢ ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8		inidm 3965	. . . . . . 7 ⊢ (𝑣 ∩ 𝑣) = 𝑣
9		eqidd 2761	. . . . . . 7 ⊢ (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖 ∈ 𝑣) → (𝑠‘𝑖) = (𝑠‘𝑖))
10		fvresi 6604	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖)
11	10	adantl 473	. . . . . . 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 7070	. . . . . 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 2766	. . . . 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 6830	. . . 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 6886	. . 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 4893	. 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 2782	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 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  𝒫 cpw 4302   ↦ cmpt 4881   I cid 5173   ↾ cres 5268   Fn wfn 6044  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816   ∘_𝑓 cof 7061   ↑_𝑚 cmap 8025  Basecbs 16079  Scalarcsca 16166   ·_𝑠 cvsca 16167   Σ_g cgsu 16323   linC clinc 42721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-1st 7334  df-2nd 7335  df-map 8027  df-linc 42723

This theorem is referenced by: (None)