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Theorem lcoop 46387
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 3462 . . 3 (𝑀𝑋𝑀 ∈ V)
21adantr 482 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ V)
3 lcoop.b . . . . . 6 𝐵 = (Base‘𝑀)
43pweqi 4575 . . . . 5 𝒫 𝐵 = 𝒫 (Base‘𝑀)
54eleq2i 2830 . . . 4 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
65biimpi 215 . . 3 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
76adantl 483 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
83fvexi 6854 . . 3 𝐵 ∈ V
9 rabexg 5287 . . 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 6840 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
1211, 3eqtr4di 2796 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
1312adantr 482 . . . 4 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘𝑚) = 𝐵)
14 2fveq3 6845 . . . . . . . 8 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
1514adantr 482 . . . . . . 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 6843 . . . . . . . 8 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
1916, 18eqtri 2766 . . . . . . 7 𝑅 = (Base‘(Scalar‘𝑀))
2015, 19eqtr4di 2796 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = 𝑅)
21 simpr 486 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑣 = 𝑉)
2220, 21oveq12d 7370 . . . . 5 ((𝑚 = 𝑀𝑣 = 𝑉) → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = (𝑅m 𝑉))
23 2fveq3 6845 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑀)))
2417a1i 11 . . . . . . . . . . 11 (𝑚 = 𝑀𝑆 = (Scalar‘𝑀))
2524eqcomd 2744 . . . . . . . . . 10 (𝑚 = 𝑀 → (Scalar‘𝑀) = 𝑆)
2625fveq2d 6844 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑀)) = (0g𝑆))
2723, 26eqtrd 2778 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑆))
2827adantr 482 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (0g‘(Scalar‘𝑚)) = (0g𝑆))
2928breq2d 5116 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑠 finSupp (0g𝑆)))
30 fveq2 6840 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
3130adantr 482 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → ( linC ‘𝑚) = ( linC ‘𝑀))
32 eqidd 2739 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑠 = 𝑠)
3331, 32, 21oveq123d 7373 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠( linC ‘𝑚)𝑣) = (𝑠( linC ‘𝑀)𝑉))
3433eqeq2d 2749 . . . . . 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 3319 . . . 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 3423 . . 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 4576 . . 3 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
39 df-lco 46383 . . 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 7503 . 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 397   = wceq 1542  wcel 2107  wrex 3072  {crab 3406  Vcvv 3444  𝒫 cpw 4559   class class class wbr 5104  cfv 6494  (class class class)co 7352  m cmap 8724   finSupp cfsupp 9264  Basecbs 17043  Scalarcsca 17096  0gc0g 17281   linC clinc 46380   LinCo clinco 46381
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6446  df-fun 6496  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-lco 46383
This theorem is referenced by:  lcoval  46388  lco0  46403
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