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Theorem lcoop 48333
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 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 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.
StepHypRef Expression
1 elex 3500 . . 3 (𝑀𝑋𝑀 ∈ V)
21adantr 480 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ V)
3 lcoop.b . . . . . 6 𝐵 = (Base‘𝑀)
43pweqi 4615 . . . . 5 𝒫 𝐵 = 𝒫 (Base‘𝑀)
54eleq2i 2832 . . . 4 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
65biimpi 216 . . 3 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
76adantl 481 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
83fvexi 6919 . . 3 𝐵 ∈ V
9 rabexg 5336 . . 3 (𝐵 ∈ V → {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V)
108, 9mp1i 13 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V)
11 fveq2 6905 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
1211, 3eqtr4di 2794 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
1312adantr 480 . . . 4 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘𝑚) = 𝐵)
14 2fveq3 6910 . . . . . . . 8 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
1514adantr 480 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
16 lcoop.r . . . . . . . 8 𝑅 = (Base‘𝑆)
17 lcoop.s . . . . . . . . 9 𝑆 = (Scalar‘𝑀)
1817fveq2i 6908 . . . . . . . 8 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
1916, 18eqtri 2764 . . . . . . 7 𝑅 = (Base‘(Scalar‘𝑀))
2015, 19eqtr4di 2794 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = 𝑅)
21 simpr 484 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑣 = 𝑉)
2220, 21oveq12d 7450 . . . . 5 ((𝑚 = 𝑀𝑣 = 𝑉) → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = (𝑅m 𝑉))
23 2fveq3 6910 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑀)))
2417a1i 11 . . . . . . . . . . 11 (𝑚 = 𝑀𝑆 = (Scalar‘𝑀))
2524eqcomd 2742 . . . . . . . . . 10 (𝑚 = 𝑀 → (Scalar‘𝑀) = 𝑆)
2625fveq2d 6909 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑀)) = (0g𝑆))
2723, 26eqtrd 2776 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑆))
2827adantr 480 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (0g‘(Scalar‘𝑚)) = (0g𝑆))
2928breq2d 5154 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑠 finSupp (0g𝑆)))
30 fveq2 6905 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
3130adantr 480 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → ( linC ‘𝑚) = ( linC ‘𝑀))
32 eqidd 2737 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑠 = 𝑠)
3331, 32, 21oveq123d 7453 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠( linC ‘𝑚)𝑣) = (𝑠( linC ‘𝑀)𝑉))
3433eqeq2d 2747 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑐 = (𝑠( linC ‘𝑚)𝑣) ↔ 𝑐 = (𝑠( linC ‘𝑀)𝑉)))
3529, 34anbi12d 632 . . . . 5 ((𝑚 = 𝑀𝑣 = 𝑉) → ((𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣)) ↔ (𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))))
3622, 35rexeqbidv 3346 . . . 4 ((𝑚 = 𝑀𝑣 = 𝑉) → (∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣)) ↔ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))))
3713, 36rabeqbidv 3454 . . 3 ((𝑚 = 𝑀𝑣 = 𝑉) → {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))} = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
3811pweqd 4616 . . 3 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
39 df-lco 48329 . . 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))})
4037, 38, 39ovmpox 7587 . 2 ((𝑀 ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
412, 7, 10, 40syl3anc 1372 1 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wrex 3069  {crab 3435  Vcvv 3479  𝒫 cpw 4599   class class class wbr 5142  cfv 6560  (class class class)co 7432  m cmap 8867   finSupp cfsupp 9402  Basecbs 17248  Scalarcsca 17301  0gc0g 17485   linC clinc 48326   LinCo clinco 48327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-lco 48329
This theorem is referenced by:  lcoval  48334  lco0  48349
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