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Theorem lcoval 45287
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 45286 . . 3 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
54eleq2d 2818 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ 𝐶 ∈ {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))}))
6 eqeq1 2742 . . . . 5 (𝑐 = 𝐶 → (𝑐 = (𝑠( linC ‘𝑀)𝑉) ↔ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))
76anbi2d 632 . . . 4 (𝑐 = 𝐶 → ((𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ (𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
87rexbidv 3207 . . 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 3588 . 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 290 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 209  wa 399   = wceq 1542  wcel 2114  wrex 3054  {crab 3057  𝒫 cpw 4488   class class class wbr 5030  cfv 6339  (class class class)co 7170  m cmap 8437   finSupp cfsupp 8906  Basecbs 16586  Scalarcsca 16671  0gc0g 16816   linC clinc 45279   LinCo clinco 45280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-lco 45282
This theorem is referenced by:  lcoel0  45303  lincsumcl  45306  lincscmcl  45307  lincolss  45309  ellcoellss  45310  lcoss  45311  lindslinindsimp1  45332  lindslinindsimp2  45338
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