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Theorem lcoop 42801
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 3364 . . 3 (𝑀𝑋𝑀 ∈ V)
21adantr 472 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ V)
3 lcoop.b . . . . . 6 𝐵 = (Base‘𝑀)
43pweqi 4318 . . . . 5 𝒫 𝐵 = 𝒫 (Base‘𝑀)
54eleq2i 2835 . . . 4 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
65biimpi 207 . . 3 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
76adantl 473 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
83fvexi 6388 . . 3 𝐵 ∈ V
9 rabexg 4971 . . 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 6374 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
1211, 3syl6eqr 2816 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
1312adantr 472 . . . 4 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘𝑚) = 𝐵)
14 2fveq3 6379 . . . . . . . 8 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
1514adantr 472 . . . . . . 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 6377 . . . . . . . 8 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
1916, 18eqtri 2786 . . . . . . 7 𝑅 = (Base‘(Scalar‘𝑀))
2015, 19syl6eqr 2816 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = 𝑅)
21 simpr 477 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑣 = 𝑉)
2220, 21oveq12d 6859 . . . . 5 ((𝑚 = 𝑀𝑣 = 𝑉) → ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) = (𝑅𝑚 𝑉))
23 2fveq3 6379 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑀)))
2417a1i 11 . . . . . . . . . . 11 (𝑚 = 𝑀𝑆 = (Scalar‘𝑀))
2524eqcomd 2770 . . . . . . . . . 10 (𝑚 = 𝑀 → (Scalar‘𝑀) = 𝑆)
2625fveq2d 6378 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑀)) = (0g𝑆))
2723, 26eqtrd 2798 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑆))
2827adantr 472 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (0g‘(Scalar‘𝑚)) = (0g𝑆))
2928breq2d 4820 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑠 finSupp (0g𝑆)))
30 fveq2 6374 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
3130adantr 472 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → ( linC ‘𝑚) = ( linC ‘𝑀))
32 eqidd 2765 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑠 = 𝑠)
3331, 32, 21oveq123d 6862 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠( linC ‘𝑚)𝑣) = (𝑠( linC ‘𝑀)𝑉))
3433eqeq2d 2774 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑐 = (𝑠( linC ‘𝑚)𝑣) ↔ 𝑐 = (𝑠( linC ‘𝑀)𝑉)))
3529, 34anbi12d 624 . . . . 5 ((𝑚 = 𝑀𝑣 = 𝑉) → ((𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣)) ↔ (𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))))
3622, 35rexeqbidv 3300 . . . 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 3343 . . 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 4319 . . 3 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
39 df-lco 42797 . . 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, 39ovmpt2x 6986 . 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 1490 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 384   = wceq 1652  wcel 2155  wrex 3055  {crab 3058  Vcvv 3349  𝒫 cpw 4314   class class class wbr 4808  cfv 6067  (class class class)co 6841  𝑚 cmap 8059   finSupp cfsupp 8481  Basecbs 16131  Scalarcsca 16218  0gc0g 16367   linC clinc 42794   LinCo clinco 42795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-iota 6030  df-fun 6069  df-fv 6075  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-lco 42797
This theorem is referenced by:  lcoval  42802  lco0  42817
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