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Theorem lcoval 48329
Description: The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lcoop.b 𝐵 = (Base‘𝑀)
lcoop.s 𝑆 = (Scalar‘𝑀)
lcoop.r 𝑅 = (Base‘𝑆)
Assertion
Ref Expression
lcoval ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶𝐵 ∧ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))
Distinct variable groups:   𝑀,𝑠   𝑅,𝑠   𝑉,𝑠   𝐶,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝑆(𝑠)   𝑋(𝑠)

Proof of Theorem lcoval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 lcoop.b . . . 4 𝐵 = (Base‘𝑀)
2 lcoop.s . . . 4 𝑆 = (Scalar‘𝑀)
3 lcoop.r . . . 4 𝑅 = (Base‘𝑆)
41, 2, 3lcoop 48328 . . 3 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
54eleq2d 2827 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ 𝐶 ∈ {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))}))
6 eqeq1 2741 . . . . 5 (𝑐 = 𝐶 → (𝑐 = (𝑠( linC ‘𝑀)𝑉) ↔ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))
76anbi2d 630 . . . 4 (𝑐 = 𝐶 → ((𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ (𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
87rexbidv 3179 . . 3 (𝑐 = 𝐶 → (∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
98elrab 3692 . 2 (𝐶 ∈ {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ↔ (𝐶𝐵 ∧ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
105, 9bitrdi 287 1 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶𝐵 ∧ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  𝒫 cpw 4600   class class class wbr 5143  cfv 6561  (class class class)co 7431  m cmap 8866   finSupp cfsupp 9401  Basecbs 17247  Scalarcsca 17300  0gc0g 17484   linC clinc 48321   LinCo clinco 48322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-lco 48324
This theorem is referenced by:  lcoel0  48345  lincsumcl  48348  lincscmcl  48349  lincolss  48351  ellcoellss  48352  lcoss  48353  lindslinindsimp1  48374  lindslinindsimp2  48380
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