Theorem lcoop 48400

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 3468	. . 3 ⊢ (𝑀 ∈ 𝑋 → 𝑀 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ V)
3		lcoop.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
4	3	pweqi 4579	. . . . 5 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
5	4	eleq2i 2820	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
6	5	biimpi 216	. . 3 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
7	6	adantl 481	. 2 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
8	3	fvexi 6872	. . 3 ⊢ 𝐵 ∈ V
9		rabexg 5292	. . 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 6858	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
12	11, 3	eqtr4di 2782	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
13	12	adantr 480	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (Base‘𝑚) = 𝐵)
14		2fveq3 6863	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
15	14	adantr 480	. . . . . . 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 6861	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀))
19	16, 18	eqtri 2752	. . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
20	15, 19	eqtr4di 2782	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = 𝑅)
21		simpr 484	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → 𝑣 = 𝑉)
22	20, 21	oveq12d 7405	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = (𝑅 ↑m 𝑉))
23		2fveq3 6863	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑀)))
24	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → 𝑆 = (Scalar‘𝑀))
25	24	eqcomd 2735	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (Scalar‘𝑀) = 𝑆)
26	25	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑀)) = (0g‘𝑆))
27	23, 26	eqtrd 2764	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘𝑆))
28	27	adantr 480	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (0g‘(Scalar‘𝑚)) = (0g‘𝑆))
29	28	breq2d 5119	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑠 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑠 finSupp (0g‘𝑆)))
30		fveq2 6858	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → ( linC ‘𝑚) = ( linC ‘𝑀))
32		eqidd 2730	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → 𝑠 = 𝑠)
33	31, 32, 21	oveq123d 7408	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑠( linC ‘𝑚)𝑣) = (𝑠( linC ‘𝑀)𝑉))
34	33	eqeq2d 2740	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = 𝑉) → (𝑐 = (𝑠( linC ‘𝑚)𝑣) ↔ 𝑐 = (𝑠( linC ‘𝑀)𝑉)))
35	29, 34	anbi12d 632	. . . . 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 3320	. . . 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 3424	. . 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 4580	. . 3 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
39		df-lco 48396	. . 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 7542	. 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 1373	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 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3405  Vcvv 3447  𝒫 cpw 4563   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799   finSupp cfsupp 9312  Basecbs 17179  Scalarcsca 17223  0_gc0g 17402   linC clinc 48393   LinCo clinco 48394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-lco 48396

This theorem is referenced by:  lcoval  48401  lco0  48416