Theorem lincop 49073

Description: A linear combination as operation. (Contributed by AV, 30-Mar-2019.)

Assertion

Ref	Expression
lincop	⊢ (𝑀 ∈ 𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))

Distinct variable groups:   𝑀,𝑠,𝑣,𝑥   𝑣,𝑋

Allowed substitution hints:   𝑋(𝑥,𝑠)

Proof of Theorem lincop

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-linc 49071	. 2 ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)))))
2		2fveq3 6887	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
3	2	oveq1d 7426	. . 3 ⊢ (𝑚 = 𝑀 → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑣))
4		fveq2 6882	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5	4	pweqd 4584	. . 3 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
6		id 23	. . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
7		fveq2 6882	. . . . . 6 ⊢ (𝑚 = 𝑀 → ( ·𝑠 ‘𝑚) = ( ·𝑠 ‘𝑀))
8	7	oveqd 7428	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥) = ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))
9	8	mpteq2dv 5209	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)) = (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
10	6, 9	oveq12d 7429	. . 3 ⊢ (𝑚 = 𝑀 → (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥))) = (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
11	3, 5, 10	mpoeq123dv 7486	. 2 ⊢ (𝑚 = 𝑀 → (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
12		elex 3484	. 2 ⊢ (𝑀 ∈ 𝑋 → 𝑀 ∈ V)
13		fvex 6895	. . . 4 ⊢ (Base‘𝑀) ∈ V
14	13	pwex 5352	. . 3 ⊢ 𝒫 (Base‘𝑀) ∈ V
15		ovexd 7446	. . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
16	15	ralrimivw 3167	. . 3 ⊢ (𝑀 ∈ 𝑋 → ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
17		eqid 2769	. . . 4 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
18	17	mpoexxg2 49003	. . 3 ⊢ ((𝒫 (Base‘𝑀) ∈ V ∧ ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V) → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))) ∈ V)
19	14, 16, 18	sylancr 598	. 2 ⊢ (𝑀 ∈ 𝑋 → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))) ∈ V)
20	1, 11, 12, 19	fvmptd3 7014	1 ⊢ (𝑀 ∈ 𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463  𝒫 cpw 4567   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8824  Basecbs 17269  Scalarcsca 17313   ·_𝑠 cvsca 17314   Σ_g cgsu 17493   linC clinc 49069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-linc 49071

This theorem is referenced by:  lincval  49074