Theorem lincval 48897

Description: The value of a linear combination. (Contributed by AV, 30-Mar-2019.)

Assertion

Ref	Expression
lincval	⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))))

Distinct variable groups:   𝑥,𝑀   𝑥,𝑆   𝑥,𝑉

Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem lincval

Dummy variables 𝑠 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lincop 48896	. . . 4 ⊢ (𝑀 ∈ 𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
2	1	3ad2ant1 1134	. . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
3	2	oveqd 7377	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))))𝑉))
4		simp2 1138	. . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
5		simp3 1139	. . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
6		ovexd 7395	. . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))) ∈ V)
7		simpr 484	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑣 = 𝑉) → 𝑣 = 𝑉)
8		fveq1 6833	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑥) = (𝑆‘𝑥))
9	8	oveq1d 7375	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))
10	9	adantr 480	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑣 = 𝑉) → ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))
11	7, 10	mpteq12dv 5173	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑣 = 𝑉) → (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
12	11	oveq2d 7376	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑣 = 𝑉) → (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
13		oveq2 7368	. . . 4 ⊢ (𝑣 = 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
14		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))))
15	12, 13, 14	ovmpox2 48829	. . 3 ⊢ ((𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))) ∈ V) → (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))))𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
16	4, 5, 6, 15	syl3anc 1374	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))))𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
17	3, 16	eqtrd 2772	1 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430  𝒫 cpw 4542   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362   ↑_m cmap 8766  Basecbs 17170  Scalarcsca 17214   ·_𝑠 cvsca 17215   Σ_g cgsu 17394   linC clinc 48892

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-linc 48894

This theorem is referenced by:  lincfsuppcl  48901  linccl  48902  lincval0  48903  lincvalsng  48904  lincvalpr  48906  lincvalsc0  48909  linc0scn0  48911  lincdifsn  48912  linc1  48913  lincellss  48914  lincsum  48917  lincscm  48918  lindslinindimp2lem4  48949  lindslinindsimp2lem5  48950  snlindsntor  48959  lincresunit3lem2  48968  lincresunit3  48969  zlmodzxzldeplem3  48990