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Theorem lsmfval 19671
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 3499 . . 3 (𝐺𝑉𝐺 ∈ V)
3 fveq2 6907 . . . . . . 7 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 lsmfval.v . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2793 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
65pweqd 4622 . . . . 5 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7 fveq2 6907 . . . . . . . . 9 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
8 lsmfval.a . . . . . . . . 9 + = (+g𝐺)
97, 8eqtr4di 2793 . . . . . . . 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 5952 . . . . 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 19669 . . . 4 LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))
154fvexi 6921 . . . . . 6 𝐵 ∈ V
1615pwex 5386 . . . . 5 𝒫 𝐵 ∈ V
1716, 16mpoex 8103 . . . 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 2787 1 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  𝒫 cpw 4605  ran crn 5690  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  +gcplusg 17298  LSSumclsm 19667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-lsm 19669
This theorem is referenced by:  lsmvalx  19672  oppglsm  19675  lsmpropd  19710  rlmlsm  21230
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