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Theorem lsmfval 18759
 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 3459 . . 3 (𝐺𝑉𝐺 ∈ V)
3 fveq2 6646 . . . . . . 7 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 lsmfval.v . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2851 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
65pweqd 4516 . . . . 5 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7 fveq2 6646 . . . . . . . . 9 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
8 lsmfval.a . . . . . . . . 9 + = (+g𝐺)
97, 8eqtr4di 2851 . . . . . . . 8 (𝑤 = 𝐺 → (+g𝑤) = + )
109oveqd 7153 . . . . . . 7 (𝑤 = 𝐺 → (𝑥(+g𝑤)𝑦) = (𝑥 + 𝑦))
1110mpoeq3dv 7213 . . . . . 6 (𝑤 = 𝐺 → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
1211rneqd 5773 . . . . 5 (𝑤 = 𝐺 → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
136, 6, 12mpoeq123dv 7209 . . . 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 18757 . . . 4 LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))
154fvexi 6660 . . . . . 6 𝐵 ∈ V
1615pwex 5247 . . . . 5 𝒫 𝐵 ∈ V
1716, 16mpoex 7763 . . . 4 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))) ∈ V
1813, 14, 17fvmpt 6746 . . 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, 19syl5eq 2845 1 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441  𝒫 cpw 4497  ran crn 5521  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  Basecbs 16478  +gcplusg 16560  LSSumclsm 18755 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-lsm 18757 This theorem is referenced by:  lsmvalx  18760  oppglsm  18763  lsmpropd  18799  rlmlsm  19976
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