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Theorem lincval 46736
Description: The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
lincval ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)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.
StepHypRef Expression
1 lincop 46735 . . . 4 (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
213ad2ant1 1133 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
32oveqd 7410 . 2 ((𝑀𝑋𝑆 ∈ ((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 (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))𝑉))
4 simp2 1137 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
5 simp3 1138 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
6 ovexd 7428 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))) ∈ V)
7 simpr 485 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → 𝑣 = 𝑉)
8 fveq1 6877 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠𝑥) = (𝑆𝑥))
98oveq1d 7408 . . . . . . 7 (𝑠 = 𝑆 → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
109adantr 481 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
117, 10mpteq12dv 5232 . . . . 5 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥)))
1211oveq2d 7409 . . . 4 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
13 oveq2 7401 . . . 4 (𝑣 = 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
14 eqid 2731 . . . 4 (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
1512, 13, 14ovmpox2 46662 . . 3 ((𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))) ∈ V) → (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
164, 5, 6, 15syl3anc 1371 . 2 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
173, 16eqtrd 2771 1 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3473  𝒫 cpw 4596  cmpt 5224  cfv 6532  (class class class)co 7393  cmpo 7395  m cmap 8803  Basecbs 17126  Scalarcsca 17182   ·𝑠 cvsca 17183   Σg cgsu 17368   linC clinc 46731
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-linc 46733
This theorem is referenced by:  lincfsuppcl  46740  linccl  46741  lincval0  46742  lincvalsng  46743  lincvalpr  46745  lincvalsc0  46748  linc0scn0  46750  lincdifsn  46751  linc1  46752  lincellss  46753  lincsum  46756  lincscm  46757  lindslinindimp2lem4  46788  lindslinindsimp2lem5  46789  snlindsntor  46798  lincresunit3lem2  46807  lincresunit3  46808  zlmodzxzldeplem3  46829
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