Theorem lsmfval 18763

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	⊢ + = (+g‘𝐺)
lsmfval.s	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
lsmfval	⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (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.

Step	Hyp	Ref	Expression
1		lsmfval.s	. 2 ⊢ ⊕ = (LSSum‘𝐺)
2		elex 3512	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		fveq2 6670	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		lsmfval.v	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	syl6eqr 2874	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6	5	pweqd 4558	. . . . 5 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7		fveq2 6670	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
8		lsmfval.a	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
9	7, 8	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
10	9	oveqd 7173	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (𝑥(+g‘𝑤)𝑦) = (𝑥 + 𝑦))
11	10	mpoeq3dv 7233	. . . . . 6 ⊢ (𝑤 = 𝐺 → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))
12	11	rneqd 5808	. . . . 5 ⊢ (𝑤 = 𝐺 → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))
13	6, 6, 12	mpoeq123dv 7229	. . . 4 ⊢ (𝑤 = 𝐺 → (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))
14		df-lsm 18761	. . . 4 ⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))))
15	4	fvexi 6684	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	pwex 5281	. . . . 5 ⊢ 𝒫 𝐵 ∈ V
17	16, 16	mpoex 7777	. . . 4 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) ∈ V
18	13, 14, 17	fvmpt 6768	. . 3 ⊢ (𝐺 ∈ V → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))
19	2, 18	syl 17	. 2 ⊢ (𝐺 ∈ 𝑉 → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))
20	1, 19	syl5eq 2868	1 ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  𝒫 cpw 4539  ran crn 5556  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16483  +_gcplusg 16565  LSSumclsm 18759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-lsm 18761

This theorem is referenced by:  lsmvalx  18764  oppglsm  18767  lsmpropd  18803  rlmlsm  19979