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Theorem lincval 44298
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 44297 . . . 4 (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
213ad2ant1 1127 . . 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 7168 . 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 1131 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
5 simp3 1132 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
6 ovexd 7186 . . 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 6665 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠𝑥) = (𝑆𝑥))
98oveq1d 7166 . . . . . . 7 (𝑠 = 𝑆 → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
109adantr 481 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
117, 10mpteq12dv 5147 . . . . 5 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥)))
1211oveq2d 7167 . . . 4 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
13 oveq2 7159 . . . 4 (𝑣 = 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
14 eqid 2825 . . . 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 44223 . . 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 1365 . 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 2860 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 1081   = wceq 1530  wcel 2107  Vcvv 3499  𝒫 cpw 4541  cmpt 5142  cfv 6351  (class class class)co 7151  cmpo 7153  m cmap 8399  Basecbs 16475  Scalarcsca 16560   ·𝑠 cvsca 16561   Σg cgsu 16706   linC clinc 44293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-linc 44295
This theorem is referenced by:  lincfsuppcl  44302  linccl  44303  lincval0  44304  lincvalsng  44305  lincvalpr  44307  lincvalsc0  44310  linc0scn0  44312  lincdifsn  44313  linc1  44314  lincellss  44315  lincsum  44318  lincscm  44319  lindslinindimp2lem4  44350  lindslinindsimp2lem5  44351  snlindsntor  44360  lincresunit3lem2  44369  lincresunit3  44370  zlmodzxzldeplem3  44391
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