Theorem lsmfval 18053

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 3212	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		fveq2 6191	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		lsmfval.v	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	syl6eqr 2674	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6	5	pweqd 4163	. . . . 5 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
7		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
8		lsmfval.a	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
9	7, 8	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
10	9	oveqd 6667	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (𝑥(+g‘𝑤)𝑦) = (𝑥 + 𝑦))
11	10	mpt2eq3dv 6721	. . . . . 6 ⊢ (𝑤 = 𝐺 → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))
12	11	rneqd 5353	. . . . 5 ⊢ (𝑤 = 𝐺 → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))
13	6, 6, 12	mpt2eq123dv 6717	. . . 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 18051	. . . 4 ⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))))
15		fvex 6201	. . . . . . 7 ⊢ (Base‘𝐺) ∈ V
16	4, 15	eqeltri 2697	. . . . . 6 ⊢ 𝐵 ∈ V
17	16	pwex 4848	. . . . 5 ⊢ 𝒫 𝐵 ∈ V
18	17, 17	mpt2ex 7247	. . . 4 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) ∈ V
19	13, 14, 18	fvmpt 6282	. . 3 ⊢ (𝐺 ∈ V → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))
20	2, 19	syl 17	. 2 ⊢ (𝐺 ∈ 𝑉 → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))
21	1, 20	syl5eq 2668	1 ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  𝒫 cpw 4158  ran crn 5115  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  +_gcplusg 15941  LSSumclsm 18049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-lsm 18051

This theorem is referenced by:  lsmvalx  18054  oppglsm  18057  lsmpropd  18090