Theorem gsumfsum 14923

Description: Relate a group sum on ℂ_fld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)

Hypotheses

Ref	Expression
gsumfsum.1	⊢ (𝜑 → 𝐴 ∈ Fin)
gsumfsum.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
gsumfsum	⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)

Distinct variable groups:   𝐴,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem gsumfsum

Dummy variables 𝑤 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq1 4215	. . . 4 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
2	1	oveq2d 6101	. . 3 ⊢ (𝑤 = ∅ → (ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ ∅ ↦ 𝐵)))
3		sumeq1 12121	. . 3 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4	2, 3	eqeq12d 2253	. 2 ⊢ (𝑤 = ∅ → ((ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = Σ𝑘 ∈ 𝑤 𝐵 ↔ (ℂfld Σg (𝑘 ∈ ∅ ↦ 𝐵)) = Σ𝑘 ∈ ∅ 𝐵))
5		mpteq1 4215	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑘 ∈ 𝑤 ↦ 𝐵) = (𝑘 ∈ 𝑦 ↦ 𝐵))
6	5	oveq2d 6101	. . 3 ⊢ (𝑤 = 𝑦 → (ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)))
7		sumeq1 12121	. . 3 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
8	6, 7	eqeq12d 2253	. 2 ⊢ (𝑤 = 𝑦 → ((ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = Σ𝑘 ∈ 𝑤 𝐵 ↔ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵))
9		mpteq1 4215	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑘 ∈ 𝑤 ↦ 𝐵) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵))
10	9	oveq2d 6101	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)))
11		sumeq1 12121	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
12	10, 11	eqeq12d 2253	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = Σ𝑘 ∈ 𝑤 𝐵 ↔ (ℂfld Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
13		mpteq1 4215	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑘 ∈ 𝑤 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵))
14	13	oveq2d 6101	. . 3 ⊢ (𝑤 = 𝐴 → (ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)))
15		sumeq1 12121	. . 3 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
16	14, 15	eqeq12d 2253	. 2 ⊢ (𝑤 = 𝐴 → ((ℂfld Σg (𝑘 ∈ 𝑤 ↦ 𝐵)) = Σ𝑘 ∈ 𝑤 𝐵 ↔ (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵))
17		cnfld0 14908	. . . 4 ⊢ 0 = (0g‘ℂfld)
18		sum0 12155	. . . 4 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
19		mpt0 5511	. . . . . 6 ⊢ (𝑘 ∈ ∅ ↦ 𝐵) = ∅
20	19	oveq2i 6096	. . . . 5 ⊢ (ℂfld Σg (𝑘 ∈ ∅ ↦ 𝐵)) = (ℂfld Σg ∅)
21		cnring 14907	. . . . . . 7 ⊢ ℂfld ∈ Ring
22		ringcmn 14338	. . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
23	21, 22	ax-mp 5	. . . . . 6 ⊢ ℂfld ∈ CMnd
24		gsum0cmn 14154	. . . . . 6 ⊢ (ℂfld ∈ CMnd → (ℂfld Σg ∅) = (0g‘ℂfld))
25	23, 24	ax-mp 5	. . . . 5 ⊢ (ℂfld Σg ∅) = (0g‘ℂfld)
26	20, 25	eqtri 2259	. . . 4 ⊢ (ℂfld Σg (𝑘 ∈ ∅ ↦ 𝐵)) = (0g‘ℂfld)
27	17, 18, 26	3eqtr4ri 2270	. . 3 ⊢ (ℂfld Σg (𝑘 ∈ ∅ ↦ 𝐵)) = Σ𝑘 ∈ ∅ 𝐵
28	27	a1i 9	. 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ ∅ ↦ 𝐵)) = Σ𝑘 ∈ ∅ 𝐵)
29		simplr 533	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
30		simplll 539	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
31		simprl 535	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ⊆ 𝐴)
32	31	sseld 3247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴))
33	32	imp 124	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
34		gsumfsum.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
35	30, 33, 34	syl2anc 415	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
36	29, 35	fsumcl 12167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
37	36	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
38		simprr 537	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
39	38	eldifad 3231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
40	34	ralrimiva 2623	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
41	40	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
42		rspcsbela 3207	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
43	39, 41, 42	syl2anc 415	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
44	43	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
45	37, 44	addcld 8345	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
46		oveq1 6092	. . . . . 6 ⊢ (𝑢 = Σ𝑘 ∈ 𝑦 𝐵 → (𝑢 + 𝑣) = (Σ𝑘 ∈ 𝑦 𝐵 + 𝑣))
47		oveq2 6093	. . . . . 6 ⊢ (𝑣 = ⦋𝑧 / 𝑘⦌𝐵 → (Σ𝑘 ∈ 𝑦 𝐵 + 𝑣) = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
48		eqid 2238	. . . . . 6 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))
49	46, 47, 48	ovmpog 6223	. . . . 5 ⊢ ((Σ𝑘 ∈ 𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ ∧ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ) → (Σ𝑘 ∈ 𝑦 𝐵(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⦋𝑧 / 𝑘⦌𝐵) = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
50	37, 44, 45, 49	syl3anc 1278	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (Σ𝑘 ∈ 𝑦 𝐵(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⦋𝑧 / 𝑘⦌𝐵) = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
51		cnfldbas 14897	. . . . . . 7 ⊢ ℂ = (Base‘ℂfld)
52		mpocnfldadd 14898	. . . . . . 7 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) = (+g‘ℂfld)
53	23	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ℂfld ∈ CMnd)
54		simplll 539	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
55		elun 3370	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}))
56		elsni 3727	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {𝑧} → 𝑘 = 𝑧)
57	56	eleq1d 2307	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑧} → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
58	39, 57	syl5ibrcom 157	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ {𝑧} → 𝑘 ∈ 𝐴))
59	32, 58	jaod 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}) → 𝑘 ∈ 𝐴))
60	55, 59	biimtrid 152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ (𝑦 ∪ {𝑧}) → 𝑘 ∈ 𝐴))
61	60	imp 124	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴)
62	54, 61, 34	syl2anc 415	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℂ)
63	62	fmpttd 5863	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵):(𝑦 ∪ {𝑧})⟶ℂ)
64	38	eldifbd 3232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
65	51, 52, 53, 63, 29, 39, 64	gsump1 14157	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (ℂfld Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)) = ((ℂfld Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵) ↾ 𝑦))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)‘𝑧)))
66	65	adantr 276	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (ℂfld Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)) = ((ℂfld Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵) ↾ 𝑦))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)‘𝑧)))
67		ssun1 3392	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
68	67	a1i 9	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
69	68	resmptd 5114	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵) ↾ 𝑦) = (𝑘 ∈ 𝑦 ↦ 𝐵))
70	69	oveq2d 6101	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (ℂfld Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵) ↾ 𝑦)) = (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)))
71		simpr 110	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵)
72	70, 71	eqtrd 2271	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (ℂfld Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵) ↾ 𝑦)) = Σ𝑘 ∈ 𝑦 𝐵)
73		ssun2 3393	. . . . . . . 8 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
74		vsnid 3741	. . . . . . . 8 ⊢ 𝑧 ∈ {𝑧}
75	73, 74	sselii 3245	. . . . . . 7 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
76		eqid 2238	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)
77	76	fvmpts 5783	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑦 ∪ {𝑧}) ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)‘𝑧) = ⦋𝑧 / 𝑘⦌𝐵)
78	75, 44, 77	sylancr 418	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)‘𝑧) = ⦋𝑧 / 𝑘⦌𝐵)
79	72, 78	oveq12d 6103	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → ((ℂfld Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵) ↾ 𝑦))(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)‘𝑧)) = (Σ𝑘 ∈ 𝑦 𝐵(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⦋𝑧 / 𝑘⦌𝐵))
80	66, 79	eqtrd 2271	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (ℂfld Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)) = (Σ𝑘 ∈ 𝑦 𝐵(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⦋𝑧 / 𝑘⦌𝐵))
81		nfv 1581	. . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
82		nfcsb1v 3180	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
83		csbeq1a 3156	. . . . . 6 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
84	81, 82, 29, 39, 64, 35, 83, 43	fsumsplitsn 12177	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
85	84	adantr 276	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
86	50, 80, 85	3eqtr4d 2281	. . 3 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵) → (ℂfld Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
87	86	ex 115	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((ℂfld Σg (𝑘 ∈ 𝑦 ↦ 𝐵)) = Σ𝑘 ∈ 𝑦 𝐵 → (ℂfld Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 𝐵)) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
88		gsumfsum.1	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
89	4, 8, 12, 16, 28, 87, 88	findcard2sd 7196	1 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ⦋csb 3147   ∖ cdif 3217   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  {csn 3709   ↦ cmpt 4192   ↾ cres 4776  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087  Fincfn 7022  ℂcc 8177  0cc0 8179   + caddc 8182  Σcsu 12119  0_gc0g 13610  CMndccmn 14087   Σ_g cgsu 14150  Ringcrg 14300  ℂ_fldccnfld 14893

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-struct 13354  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-starv 13446  df-tset 13450  df-ple 13451  df-ds 13453  df-unif 13454  df-0g 13612  df-gzsum 13613  df-topgen 13614  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-mulg 13923  df-cmn 14089  df-abl 14090  df-gsumfi 14151  df-mgp 14218  df-ur 14263  df-ring 14302  df-cring 14303  df-bl 14883  df-mopn 14884  df-fg 14886  df-metu 14887  df-cnfld 14894

This theorem is used by:  lgseisenlem4  16192