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Theorem lsmfval 19656
Description: The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmfval.v 𝐵 = (Base‘𝐺)
lsmfval.a + = (+g𝐺)
lsmfval.s = (LSSum‘𝐺)
Assertion
Ref Expression
lsmfval (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
Distinct variable groups:   𝑢,𝑡,𝑥,𝑦, +   𝑡,𝐵,𝑢,𝑥,𝑦   𝑡,𝐺,𝑢,𝑥,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑢,𝑡)   𝑉(𝑥,𝑦,𝑢,𝑡)

Proof of Theorem lsmfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lsmfval.s . 2 = (LSSum‘𝐺)
2 elex 3501 . . 3 (𝐺𝑉𝐺 ∈ V)
3 fveq2 6906 . . . . . . 7 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 lsmfval.v . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2795 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
65pweqd 4617 . . . . 5 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7 fveq2 6906 . . . . . . . . 9 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
8 lsmfval.a . . . . . . . . 9 + = (+g𝐺)
97, 8eqtr4di 2795 . . . . . . . 8 (𝑤 = 𝐺 → (+g𝑤) = + )
109oveqd 7448 . . . . . . 7 (𝑤 = 𝐺 → (𝑥(+g𝑤)𝑦) = (𝑥 + 𝑦))
1110mpoeq3dv 7512 . . . . . 6 (𝑤 = 𝐺 → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
1211rneqd 5949 . . . . 5 (𝑤 = 𝐺 → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
136, 6, 12mpoeq123dv 7508 . . . 4 (𝑤 = 𝐺 → (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
14 df-lsm 19654 . . . 4 LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))
154fvexi 6920 . . . . . 6 𝐵 ∈ V
1615pwex 5380 . . . . 5 𝒫 𝐵 ∈ V
1716, 16mpoex 8104 . . . 4 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))) ∈ V
1813, 14, 17fvmpt 7016 . . 3 (𝐺 ∈ V → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
192, 18syl 17 . 2 (𝐺𝑉 → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
201, 19eqtrid 2789 1 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  𝒫 cpw 4600  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  LSSumclsm 19652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-lsm 19654
This theorem is referenced by:  lsmvalx  19657  oppglsm  19660  lsmpropd  19695  rlmlsm  21212
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