Theorem lincop 44456

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 44454	. 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 6670	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
3	2	oveq1d 7165	. . 3 ⊢ (𝑚 = 𝑀 → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑣))
4		fveq2 6665	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5	4	pweqd 4544	. . 3 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
6		id 22	. . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
7		fveq2 6665	. . . . . 6 ⊢ (𝑚 = 𝑀 → ( ·𝑠 ‘𝑚) = ( ·𝑠 ‘𝑀))
8	7	oveqd 7167	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥) = ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))
9	8	mpteq2dv 5155	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)) = (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
10	6, 9	oveq12d 7168	. . 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 7223	. 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 3513	. 2 ⊢ (𝑀 ∈ 𝑋 → 𝑀 ∈ V)
13		fvex 6678	. . . 4 ⊢ (Base‘𝑀) ∈ V
14	13	pwex 5274	. . 3 ⊢ 𝒫 (Base‘𝑀) ∈ V
15		ovexd 7185	. . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
16	15	ralrimivw 3183	. . 3 ⊢ (𝑀 ∈ 𝑋 → ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
17		eqid 2821	. . . 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 44379	. . 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 589	. 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 6786	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 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3495  𝒫 cpw 4539   ↦ cmpt 5139  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152   ↑_m cmap 8400  Basecbs 16477  Scalarcsca 16562   ·_𝑠 cvsca 16563   Σ_g cgsu 16708   linC clinc 44452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-linc 44454

This theorem is referenced by:  lincval  44457