Theorem lcoop 44820

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 3459	. . 3 ⊢ (𝑀 ∈ 𝑋 → 𝑀 ∈ V)
2	1	adantr 484	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ V)
3		lcoop.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
4	3	pweqi 4515	. . . . 5 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
5	4	eleq2i 2881	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
6	5	biimpi 219	. . 3 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
7	6	adantl 485	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
8	3	fvexi 6659	. . 3 ⊢ 𝐵 ∈ V
9		rabexg 5198	. . 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 6645	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
12	11, 3	eqtr4di 2851	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
13	12	adantr 484	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (Base‘𝑚) = 𝐵)
14		2fveq3 6650	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
15	14	adantr 484	. . . . . . 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 6648	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀))
19	16, 18	eqtri 2821	. . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
20	15, 19	eqtr4di 2851	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = 𝑅)
21		simpr 488	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → 𝑣 = 𝑉)
22	20, 21	oveq12d 7153	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = (𝑅 ↑m 𝑉))
23		2fveq3 6650	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑀)))
24	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → 𝑆 = (Scalar‘𝑀))
25	24	eqcomd 2804	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (Scalar‘𝑀) = 𝑆)
26	25	fveq2d 6649	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑀)) = (0g‘𝑆))
27	23, 26	eqtrd 2833	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘𝑆))
28	27	adantr 484	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (0g‘(Scalar‘𝑚)) = (0g‘𝑆))
29	28	breq2d 5042	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑠 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑠 finSupp (0g‘𝑆)))
30		fveq2 6645	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
31	30	adantr 484	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → ( linC ‘𝑚) = ( linC ‘𝑀))
32		eqidd 2799	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → 𝑠 = 𝑠)
33	31, 32, 21	oveq123d 7156	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑠( linC ‘𝑚)𝑣) = (𝑠( linC ‘𝑀)𝑉))
34	33	eqeq2d 2809	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑐 = (𝑠( linC ‘𝑚)𝑣) ↔ 𝑐 = (𝑠( linC ‘𝑀)𝑉)))
35	29, 34	anbi12d 633	. . . . 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 3355	. . . 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 3433	. . 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 4516	. . 3 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
39		df-lco 44816	. . 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 7282	. 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 1368	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 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  Vcvv 3441  𝒫 cpw 4497   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389   finSupp cfsupp 8817  Basecbs 16475  Scalarcsca 16560  0_gc0g 16705   linC clinc 44813   LinCo clinco 44814

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-lco 44816

This theorem is referenced by:  lcoval  44821  lco0  44836