Theorem lcoval 47666

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	⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 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.

Step	Hyp	Ref	Expression
1		lcoop.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
2		lcoop.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑀)
3		lcoop.r	. . . 4 ⊢ 𝑅 = (Base‘𝑆)
4	1, 2, 3	lcoop 47665	. . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
5	4	eleq2d 2811	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ 𝐶 ∈ {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))}))
6		eqeq1 2729	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 = (𝑠( linC ‘𝑀)𝑉) ↔ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))
7	6	anbi2d 628	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ (𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
8	7	rexbidv 3168	. . 3 ⊢ (𝑐 = 𝐶 → (∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
9	8	elrab 3679	. 2 ⊢ (𝐶 ∈ {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
10	5, 9	bitrdi 286	1 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059  {crab 3418  𝒫 cpw 4604   class class class wbr 5149  ‘cfv 6549  (class class class)co 7419   ↑_m cmap 8845   finSupp cfsupp 9387  Basecbs 17183  Scalarcsca 17239  0_gc0g 17424   linC clinc 47658   LinCo clinco 47659

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-lco 47661

This theorem is referenced by:  lcoel0  47682  lincsumcl  47685  lincscmcl  47686  lincolss  47688  ellcoellss  47689  lcoss  47690  lindslinindsimp1  47711  lindslinindsimp2  47717