Theorem lincop 48899

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 48897	. 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 6840	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
3	2	oveq1d 7376	. . 3 ⊢ (𝑚 = 𝑀 → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑣))
4		fveq2 6835	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5	4	pweqd 4559	. . 3 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
6		id 22	. . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
7		fveq2 6835	. . . . . 6 ⊢ (𝑚 = 𝑀 → ( ·𝑠 ‘𝑚) = ( ·𝑠 ‘𝑀))
8	7	oveqd 7378	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥) = ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))
9	8	mpteq2dv 5180	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)) = (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
10	6, 9	oveq12d 7379	. . 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 7436	. 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 3451	. 2 ⊢ (𝑀 ∈ 𝑋 → 𝑀 ∈ V)
13		fvex 6848	. . . 4 ⊢ (Base‘𝑀) ∈ V
14	13	pwex 5318	. . 3 ⊢ 𝒫 (Base‘𝑀) ∈ V
15		ovexd 7396	. . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
16	15	ralrimivw 3134	. . 3 ⊢ (𝑀 ∈ 𝑋 → ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
17		eqid 2737	. . . 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 48829	. . 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 588	. 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 6966	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 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  𝒫 cpw 4542   ↦ cmpt 5167  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363   ↑_m cmap 8767  Basecbs 17173  Scalarcsca 17217   ·_𝑠 cvsca 17218   Σ_g cgsu 17397   linC clinc 48895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-linc 48897

This theorem is referenced by:  lincval  48900