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Theorem lsmfval 19592
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 3490 . . 3 (𝐺𝑉𝐺 ∈ V)
3 fveq2 6897 . . . . . . 7 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 lsmfval.v . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2786 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
65pweqd 4620 . . . . 5 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7 fveq2 6897 . . . . . . . . 9 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
8 lsmfval.a . . . . . . . . 9 + = (+g𝐺)
97, 8eqtr4di 2786 . . . . . . . 8 (𝑤 = 𝐺 → (+g𝑤) = + )
109oveqd 7437 . . . . . . 7 (𝑤 = 𝐺 → (𝑥(+g𝑤)𝑦) = (𝑥 + 𝑦))
1110mpoeq3dv 7499 . . . . . 6 (𝑤 = 𝐺 → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
1211rneqd 5940 . . . . 5 (𝑤 = 𝐺 → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
136, 6, 12mpoeq123dv 7495 . . . 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 19590 . . . 4 LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))
154fvexi 6911 . . . . . 6 𝐵 ∈ V
1615pwex 5380 . . . . 5 𝒫 𝐵 ∈ V
1716, 16mpoex 8084 . . . 4 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))) ∈ V
1813, 14, 17fvmpt 7005 . . 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 2780 1 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3471  𝒫 cpw 4603  ran crn 5679  cfv 6548  (class class class)co 7420  cmpo 7422  Basecbs 17179  +gcplusg 17232  LSSumclsm 19588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-lsm 19590
This theorem is referenced by:  lsmvalx  19593  oppglsm  19596  lsmpropd  19631  rlmlsm  21097
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