Theorem lcoop 45752

Description: A linear combination as operation. (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
lcoop	⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})

Distinct variable groups:   𝐵,𝑐   𝑀,𝑐,𝑠   𝑅,𝑐,𝑠   𝑉,𝑐,𝑠

Allowed substitution hints:   𝐵(𝑠)   𝑆(𝑠,𝑐)   𝑋(𝑠,𝑐)

Proof of Theorem lcoop

Dummy variables 𝑚 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. . 3 ⊢ (𝑀 ∈ 𝑋 → 𝑀 ∈ V)
2	1	adantr 481	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ V)
3		lcoop.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
4	3	pweqi 4551	. . . . 5 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
5	4	eleq2i 2830	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
6	5	biimpi 215	. . 3 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
7	6	adantl 482	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
8	3	fvexi 6788	. . 3 ⊢ 𝐵 ∈ V
9		rabexg 5255	. . 3 ⊢ (𝐵 ∈ V → {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V)
10	8, 9	mp1i 13	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V)
11		fveq2 6774	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
12	11, 3	eqtr4di 2796	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
13	12	adantr 481	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (Base‘𝑚) = 𝐵)
14		2fveq3 6779	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
15	14	adantr 481	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
16		lcoop.r	. . . . . . . 8 ⊢ 𝑅 = (Base‘𝑆)
17		lcoop.s	. . . . . . . . 9 ⊢ 𝑆 = (Scalar‘𝑀)
18	17	fveq2i 6777	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀))
19	16, 18	eqtri 2766	. . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
20	15, 19	eqtr4di 2796	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = 𝑅)
21		simpr 485	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → 𝑣 = 𝑉)
22	20, 21	oveq12d 7293	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = (𝑅 ↑m 𝑉))
23		2fveq3 6779	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑀)))
24	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → 𝑆 = (Scalar‘𝑀))
25	24	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (Scalar‘𝑀) = 𝑆)
26	25	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑀)) = (0g‘𝑆))
27	23, 26	eqtrd 2778	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘𝑆))
28	27	adantr 481	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (0g‘(Scalar‘𝑚)) = (0g‘𝑆))
29	28	breq2d 5086	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑠 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑠 finSupp (0g‘𝑆)))
30		fveq2 6774	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
31	30	adantr 481	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → ( linC ‘𝑚) = ( linC ‘𝑀))
32		eqidd 2739	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → 𝑠 = 𝑠)
33	31, 32, 21	oveq123d 7296	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑠( linC ‘𝑚)𝑣) = (𝑠( linC ‘𝑀)𝑉))
34	33	eqeq2d 2749	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑐 = (𝑠( linC ‘𝑚)𝑣) ↔ 𝑐 = (𝑠( linC ‘𝑀)𝑉)))
35	29, 34	anbi12d 631	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → ((𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣)) ↔ (𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))))
36	22, 35	rexeqbidv 3337	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣)) ↔ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))))
37	13, 36	rabeqbidv 3420	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))} = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
38	11	pweqd 4552	. . 3 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
39		df-lco 45748	. . 3 ⊢ LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})
40	37, 38, 39	ovmpox 7426	. 2 ⊢ ((𝑀 ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
41	2, 7, 10, 40	syl3anc 1370	1 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {crab 3068  Vcvv 3432  𝒫 cpw 4533   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615   finSupp cfsupp 9128  Basecbs 16912  Scalarcsca 16965  0_gc0g 17150   linC clinc 45745   LinCo clinco 45746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-lco 45748

This theorem is referenced by:  lcoval  45753  lco0  45768