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Theorem lsmfval 19600
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 3492 . . 3 (𝐺𝑉𝐺 ∈ V)
3 fveq2 6902 . . . . . . 7 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 lsmfval.v . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2786 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
65pweqd 4623 . . . . 5 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7 fveq2 6902 . . . . . . . . 9 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
8 lsmfval.a . . . . . . . . 9 + = (+g𝐺)
97, 8eqtr4di 2786 . . . . . . . 8 (𝑤 = 𝐺 → (+g𝑤) = + )
109oveqd 7443 . . . . . . 7 (𝑤 = 𝐺 → (𝑥(+g𝑤)𝑦) = (𝑥 + 𝑦))
1110mpoeq3dv 7505 . . . . . 6 (𝑤 = 𝐺 → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
1211rneqd 5944 . . . . 5 (𝑤 = 𝐺 → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
136, 6, 12mpoeq123dv 7501 . . . 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 19598 . . . 4 LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))
154fvexi 6916 . . . . . 6 𝐵 ∈ V
1615pwex 5384 . . . . 5 𝒫 𝐵 ∈ V
1716, 16mpoex 8090 . . . 4 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))) ∈ V
1813, 14, 17fvmpt 7010 . . 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 1533  wcel 2098  Vcvv 3473  𝒫 cpw 4606  ran crn 5683  cfv 6553  (class class class)co 7426  cmpo 7428  Basecbs 17187  +gcplusg 17240  LSSumclsm 19596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-lsm 19598
This theorem is referenced by:  lsmvalx  19601  oppglsm  19604  lsmpropd  19639  rlmlsm  21105
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