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Theorem lincval 42992
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 42991 . . . 4 (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
213ad2ant1 1164 . . 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 6896 . 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 1168 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
5 simp3 1169 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
6 ovexd 6913 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))) ∈ V)
7 simpr 478 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → 𝑣 = 𝑉)
8 fveq1 6411 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠𝑥) = (𝑆𝑥))
98oveq1d 6894 . . . . . . 7 (𝑠 = 𝑆 → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
109adantr 473 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
117, 10mpteq12dv 4927 . . . . 5 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥)))
1211oveq2d 6895 . . . 4 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
13 oveq2 6887 . . . 4 (𝑣 = 𝑉 → ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
14 eqid 2800 . . . 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, 14ovmpt2x2 42913 . . 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 1491 . 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 2834 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 385  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3386  𝒫 cpw 4350  cmpt 4923  cfv 6102  (class class class)co 6879  cmpt2 6881  𝑚 cmap 8096  Basecbs 16183  Scalarcsca 16269   ·𝑠 cvsca 16270   Σg cgsu 16415   linC clinc 42987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-linc 42989
This theorem is referenced by:  lincfsuppcl  42996  linccl  42997  lincval0  42998  lincvalsng  42999  lincvalpr  43001  lincvalsc0  43004  linc0scn0  43006  lincdifsn  43007  linc1  43008  lincellss  43009  lincsum  43012  lincscm  43013  lindslinindimp2lem4  43044  lindslinindsimp2lem5  43045  snlindsntor  43054  lincresunit3lem2  43063  lincresunit3  43064  zlmodzxzldeplem3  43085
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