Theorem lcoval 48900

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 48899	. . 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 2823	. 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 2741	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 = (𝑠( linC ‘𝑀)𝑉) ↔ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))
7	6	anbi2d 631	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉)) ↔ (𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))
8	7	rexbidv 3162	. . 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 3635	. 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 287	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3390  𝒫 cpw 4542   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766   finSupp cfsupp 9267  Basecbs 17170  Scalarcsca 17214  0_gc0g 17393   linC clinc 48892   LinCo clinco 48893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-lco 48895

This theorem is referenced by:  lcoel0  48916  lincsumcl  48919  lincscmcl  48920  lincolss  48922  ellcoellss  48923  lcoss  48924  lindslinindsimp1  48945  lindslinindsimp2  48951