Theorem lsmfval 19027

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 3426	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		fveq2 6717	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		lsmfval.v	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2796	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6	5	pweqd 4532	. . . . 5 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7		fveq2 6717	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
8		lsmfval.a	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
10	9	oveqd 7230	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (𝑥(+g‘𝑤)𝑦) = (𝑥 + 𝑦))
11	10	mpoeq3dv 7290	. . . . . 6 ⊢ (𝑤 = 𝐺 → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))
12	11	rneqd 5807	. . . . 5 ⊢ (𝑤 = 𝐺 → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))
13	6, 6, 12	mpoeq123dv 7286	. . . 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 19025	. . . 4 ⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))))
15	4	fvexi 6731	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	pwex 5273	. . . . 5 ⊢ 𝒫 𝐵 ∈ V
17	16, 16	mpoex 7850	. . . 4 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) ∈ V
18	13, 14, 17	fvmpt 6818	. . 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 2790	1 ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  Vcvv 3408  𝒫 cpw 4513  ran crn 5552  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  Basecbs 16760  +_gcplusg 16802  LSSumclsm 19023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-lsm 19025

This theorem is referenced by:  lsmvalx  19028  oppglsm  19031  lsmpropd  19067  rlmlsm  20244