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Theorem lcoop 47082
Description: A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lcoop.b 𝐡 = (Baseβ€˜π‘€)
lcoop.s 𝑆 = (Scalarβ€˜π‘€)
lcoop.r 𝑅 = (Baseβ€˜π‘†)
Assertion
Ref Expression
lcoop ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐡 ∣ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))})
Distinct variable groups:   𝐡,𝑐   𝑀,𝑐,𝑠   𝑅,𝑐,𝑠   𝑉,𝑐,𝑠
Allowed substitution hints:   𝐡(𝑠)   𝑆(𝑠,𝑐)   𝑋(𝑠,𝑐)

Proof of Theorem lcoop
Dummy variables π‘š 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3492 . . 3 (𝑀 ∈ 𝑋 β†’ 𝑀 ∈ V)
21adantr 481 . 2 ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑀 ∈ V)
3 lcoop.b . . . . . 6 𝐡 = (Baseβ€˜π‘€)
43pweqi 4618 . . . . 5 𝒫 𝐡 = 𝒫 (Baseβ€˜π‘€)
54eleq2i 2825 . . . 4 (𝑉 ∈ 𝒫 𝐡 ↔ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
65biimpi 215 . . 3 (𝑉 ∈ 𝒫 𝐡 β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
76adantl 482 . 2 ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
83fvexi 6905 . . 3 𝐡 ∈ V
9 rabexg 5331 . . 3 (𝐡 ∈ V β†’ {𝑐 ∈ 𝐡 ∣ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))} ∈ V)
108, 9mp1i 13 . 2 ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ {𝑐 ∈ 𝐡 ∣ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))} ∈ V)
11 fveq2 6891 . . . . . 6 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = (Baseβ€˜π‘€))
1211, 3eqtr4di 2790 . . . . 5 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = 𝐡)
1312adantr 481 . . . 4 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (Baseβ€˜π‘š) = 𝐡)
14 2fveq3 6896 . . . . . . . 8 (π‘š = 𝑀 β†’ (Baseβ€˜(Scalarβ€˜π‘š)) = (Baseβ€˜(Scalarβ€˜π‘€)))
1514adantr 481 . . . . . . 7 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (Baseβ€˜(Scalarβ€˜π‘š)) = (Baseβ€˜(Scalarβ€˜π‘€)))
16 lcoop.r . . . . . . . 8 𝑅 = (Baseβ€˜π‘†)
17 lcoop.s . . . . . . . . 9 𝑆 = (Scalarβ€˜π‘€)
1817fveq2i 6894 . . . . . . . 8 (Baseβ€˜π‘†) = (Baseβ€˜(Scalarβ€˜π‘€))
1916, 18eqtri 2760 . . . . . . 7 𝑅 = (Baseβ€˜(Scalarβ€˜π‘€))
2015, 19eqtr4di 2790 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (Baseβ€˜(Scalarβ€˜π‘š)) = 𝑅)
21 simpr 485 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ 𝑣 = 𝑉)
2220, 21oveq12d 7426 . . . . 5 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) = (𝑅 ↑m 𝑉))
23 2fveq3 6896 . . . . . . . . 9 (π‘š = 𝑀 β†’ (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜(Scalarβ€˜π‘€)))
2417a1i 11 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ 𝑆 = (Scalarβ€˜π‘€))
2524eqcomd 2738 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (Scalarβ€˜π‘€) = 𝑆)
2625fveq2d 6895 . . . . . . . . 9 (π‘š = 𝑀 β†’ (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜π‘†))
2723, 26eqtrd 2772 . . . . . . . 8 (π‘š = 𝑀 β†’ (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜π‘†))
2827adantr 481 . . . . . . 7 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜π‘†))
2928breq2d 5160 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ↔ 𝑠 finSupp (0gβ€˜π‘†)))
30 fveq2 6891 . . . . . . . . 9 (π‘š = 𝑀 β†’ ( linC β€˜π‘š) = ( linC β€˜π‘€))
3130adantr 481 . . . . . . . 8 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ ( linC β€˜π‘š) = ( linC β€˜π‘€))
32 eqidd 2733 . . . . . . . 8 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ 𝑠 = 𝑠)
3331, 32, 21oveq123d 7429 . . . . . . 7 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (𝑠( linC β€˜π‘š)𝑣) = (𝑠( linC β€˜π‘€)𝑉))
3433eqeq2d 2743 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (𝑐 = (𝑠( linC β€˜π‘š)𝑣) ↔ 𝑐 = (𝑠( linC β€˜π‘€)𝑉)))
3529, 34anbi12d 631 . . . . 5 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ ((𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ 𝑐 = (𝑠( linC β€˜π‘š)𝑣)) ↔ (𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))))
3622, 35rexeqbidv 3343 . . . 4 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ 𝑐 = (𝑠( linC β€˜π‘š)𝑣)) ↔ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))))
3713, 36rabeqbidv 3449 . . 3 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ {𝑐 ∈ (Baseβ€˜π‘š) ∣ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ 𝑐 = (𝑠( linC β€˜π‘š)𝑣))} = {𝑐 ∈ 𝐡 ∣ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))})
3811pweqd 4619 . . 3 (π‘š = 𝑀 β†’ 𝒫 (Baseβ€˜π‘š) = 𝒫 (Baseβ€˜π‘€))
39 df-lco 47078 . . 3 LinCo = (π‘š ∈ V, 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ {𝑐 ∈ (Baseβ€˜π‘š) ∣ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ 𝑐 = (𝑠( linC β€˜π‘š)𝑣))})
4037, 38, 39ovmpox 7560 . 2 ((𝑀 ∈ V ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ {𝑐 ∈ 𝐡 ∣ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))} ∈ V) β†’ (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐡 ∣ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))})
412, 7, 10, 40syl3anc 1371 1 ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐡 ∣ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜π‘†) ∧ 𝑐 = (𝑠( linC β€˜π‘€)𝑉))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  Vcvv 3474  π’« cpw 4602   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819   finSupp cfsupp 9360  Basecbs 17143  Scalarcsca 17199  0gc0g 17384   linC clinc 47075   LinCo clinco 47076
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-lco 47078
This theorem is referenced by:  lcoval  47083  lco0  47098
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