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Theorem lcoop 47092
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 3493 . . 3 (𝑀 ∈ 𝑋 β†’ 𝑀 ∈ V)
21adantr 482 . 2 ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑀 ∈ V)
3 lcoop.b . . . . . 6 𝐡 = (Baseβ€˜π‘€)
43pweqi 4619 . . . . 5 𝒫 𝐡 = 𝒫 (Baseβ€˜π‘€)
54eleq2i 2826 . . . 4 (𝑉 ∈ 𝒫 𝐡 ↔ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
65biimpi 215 . . 3 (𝑉 ∈ 𝒫 𝐡 β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
76adantl 483 . 2 ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐡) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
83fvexi 6906 . . 3 𝐡 ∈ V
9 rabexg 5332 . . 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 6892 . . . . . 6 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = (Baseβ€˜π‘€))
1211, 3eqtr4di 2791 . . . . 5 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = 𝐡)
1312adantr 482 . . . 4 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (Baseβ€˜π‘š) = 𝐡)
14 2fveq3 6897 . . . . . . . 8 (π‘š = 𝑀 β†’ (Baseβ€˜(Scalarβ€˜π‘š)) = (Baseβ€˜(Scalarβ€˜π‘€)))
1514adantr 482 . . . . . . 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 6895 . . . . . . . 8 (Baseβ€˜π‘†) = (Baseβ€˜(Scalarβ€˜π‘€))
1916, 18eqtri 2761 . . . . . . 7 𝑅 = (Baseβ€˜(Scalarβ€˜π‘€))
2015, 19eqtr4di 2791 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (Baseβ€˜(Scalarβ€˜π‘š)) = 𝑅)
21 simpr 486 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ 𝑣 = 𝑉)
2220, 21oveq12d 7427 . . . . 5 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣) = (𝑅 ↑m 𝑉))
23 2fveq3 6897 . . . . . . . . 9 (π‘š = 𝑀 β†’ (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜(Scalarβ€˜π‘€)))
2417a1i 11 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ 𝑆 = (Scalarβ€˜π‘€))
2524eqcomd 2739 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (Scalarβ€˜π‘€) = 𝑆)
2625fveq2d 6896 . . . . . . . . 9 (π‘š = 𝑀 β†’ (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜π‘†))
2723, 26eqtrd 2773 . . . . . . . 8 (π‘š = 𝑀 β†’ (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜π‘†))
2827adantr 482 . . . . . . 7 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜π‘†))
2928breq2d 5161 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ↔ 𝑠 finSupp (0gβ€˜π‘†)))
30 fveq2 6892 . . . . . . . . 9 (π‘š = 𝑀 β†’ ( linC β€˜π‘š) = ( linC β€˜π‘€))
3130adantr 482 . . . . . . . 8 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ ( linC β€˜π‘š) = ( linC β€˜π‘€))
32 eqidd 2734 . . . . . . . 8 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ 𝑠 = 𝑠)
3331, 32, 21oveq123d 7430 . . . . . . 7 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (𝑠( linC β€˜π‘š)𝑣) = (𝑠( linC β€˜π‘€)𝑉))
3433eqeq2d 2744 . . . . . 6 ((π‘š = 𝑀 ∧ 𝑣 = 𝑉) β†’ (𝑐 = (𝑠( linC β€˜π‘š)𝑣) ↔ 𝑐 = (𝑠( linC β€˜π‘€)𝑉)))
3529, 34anbi12d 632 . . . . 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 3344 . . . 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 3450 . . 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 4620 . . 3 (π‘š = 𝑀 β†’ 𝒫 (Baseβ€˜π‘š) = 𝒫 (Baseβ€˜π‘€))
39 df-lco 47088 . . 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 7561 . 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 1372 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433  Vcvv 3475  π’« cpw 4603   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820   finSupp cfsupp 9361  Basecbs 17144  Scalarcsca 17200  0gc0g 17385   linC clinc 47085   LinCo clinco 47086
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-lco 47088
This theorem is referenced by:  lcoval  47093  lco0  47108
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