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Theorem lincval 45183
 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 45182 . . . 4 (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
213ad2ant1 1130 . . 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 7167 . 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 1134 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
5 simp3 1135 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
6 ovexd 7185 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))) ∈ V)
7 simpr 488 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → 𝑣 = 𝑉)
8 fveq1 6657 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠𝑥) = (𝑆𝑥))
98oveq1d 7165 . . . . . . 7 (𝑠 = 𝑆 → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
109adantr 484 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
117, 10mpteq12dv 5117 . . . . 5 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥)))
1211oveq2d 7166 . . . 4 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
13 oveq2 7158 . . . 4 (𝑣 = 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
14 eqid 2758 . . . 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 45109 . . 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 1368 . 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 2793 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409  𝒫 cpw 4494   ↦ cmpt 5112  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416  Basecbs 16541  Scalarcsca 16626   ·𝑠 cvsca 16627   Σg cgsu 16772   linC clinc 45178 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-linc 45180 This theorem is referenced by:  lincfsuppcl  45187  linccl  45188  lincval0  45189  lincvalsng  45190  lincvalpr  45192  lincvalsc0  45195  linc0scn0  45197  lincdifsn  45198  linc1  45199  lincellss  45200  lincsum  45203  lincscm  45204  lindslinindimp2lem4  45235  lindslinindsimp2lem5  45236  snlindsntor  45245  lincresunit3lem2  45254  lincresunit3  45255  zlmodzxzldeplem3  45276
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