Theorem fsumo1 15860

Description: The finite sum of eventually bounded functions (where the index set 𝐵 does not depend on 𝑥) is eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 22-May-2016.)

Hypotheses

Ref	Expression
fsumo1.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
fsumo1.2	⊢ (𝜑 → 𝐵 ∈ Fin)
fsumo1.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
fsumo1.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))

Assertion

Ref	Expression
fsumo1	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1))

Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘,𝑥   𝜑,𝑘,𝑥

Allowed substitution hints:   𝐶(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem fsumo1

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

Step	Hyp	Ref	Expression
1		ssid 4031	. 2 ⊢ 𝐵 ⊆ 𝐵
2		fsumo1.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 4034	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		sumeq1 15737	. . . . . . . . 9 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		sum0 15769	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
6	4, 5	eqtrdi 2796	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = 0)
7	6	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
8	7	eleq1d 2829	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1)))
9	3, 8	imbi12d 344	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1)) ↔ (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1))))
10	9	imbi2d 340	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1)))))
11		sseq1 4034	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
12		sumeq1 15737	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝑦 𝐶)
13	12	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
14	13	eleq1d 2829	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)))
15	11, 14	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1))))
16	15	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)))))
17		sseq1 4034	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
18		sumeq1 15737	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
19	18	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
20	19	eleq1d 2829	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))
21	17, 20	imbi12d 344	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1)) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
22	21	imbi2d 340	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
23		sseq1 4034	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
24		sumeq1 15737	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
25	24	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
26	25	eleq1d 2829	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1)))
27	23, 26	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1))))
28	27	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1)))))
29		fsumo1.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
30		0cn 11282	. . . . . 6 ⊢ 0 ∈ ℂ
31		o1const 15666	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1))
32	29, 30, 31	sylancl 585	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1))
33	32	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1)))
34		ssun1 4201	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		sstr 4017	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦 ⊆ 𝐵)
36	34, 35	mpan 689	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → 𝑦 ⊆ 𝐵)
37	36	imim1i 63	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)))
38		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑦)
39		disjsn 4736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
40	38, 39	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
41	40	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∩ {𝑧}) = ∅)
42		eqidd 2741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
43	2	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
44		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
45	43, 44	ssfid 9329	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
46	45	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
47	44	sselda 4008	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
48	47	adantlr 714	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
49		fsumo1.3	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
50	49	anass1rs 654	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
51		fsumo1.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
52	50, 51	o1mptrcl 15669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
53	52	an32s 651	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
54	53	adantllr 718	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
55	48, 54	syldan 590	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
56	41, 42, 46, 55	fsumsplit 15789	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
57		csbeq1a 3935	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑘⦌𝐶)
58		nfcv 2908	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑤𝐶
59		nfcsb1v 3946	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐶
60	57, 58, 59	cbvsum 15743	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶
61	44	unssbd 4217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
62		vex 3492	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ V
63	62	snss 4810	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
64	61, 63	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
65	64	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)
66	54	ralrimiva 3152	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
67		nfcsb1v 3946	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
68	67	nfel1 2925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
69		csbeq1a 3935	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
70	69	eleq1d 2829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
71	68, 70	rspc 3623	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
72	65, 66, 71	sylc 65	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
73		csbeq1 3924	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
74	73	sumsn 15794	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
75	65, 72, 74	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
76	60, 75	eqtrid 2792	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
77	76	oveq2d 7464	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
78	56, 77	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
79	78	mpteq2dva 5266	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
80	29	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ⊆ ℝ)
81		reex 11275	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
82	81	ssex 5339	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
83	80, 82	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ∈ V)
84		sumex 15736	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ 𝑦 𝐶 ∈ V
85	84	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑦 𝐶 ∈ V)
86		eqidd 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
87		eqidd 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
88	83, 85, 72, 86, 87	offval2 7734	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∘f + (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
89	79, 88	eqtr4d 2783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∘f + (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)))
90	89	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∘f + (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)))
91		id 22	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1))
92	51	ralrimiva 3152	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
93	92	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
94		nfcv 2908	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝐴
95	94, 67	nfmpt 5273	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)
96	95	nfel1 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝑂(1)
97	69	mpteq2dv 5268	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
98	97	eleq1d 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝑂(1)))
99	96, 98	rspc 3623	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝑂(1)))
100	64, 93, 99	sylc 65	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝑂(1))
101		o1add 15660	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1) ∧ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝑂(1)) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∘f + (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)) ∈ 𝑂(1))
102	91, 100, 101	syl2anr 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∘f + (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)) ∈ 𝑂(1))
103	90, 102	eqeltrd 2844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))
104	103	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))
105	104	expr 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
106	105	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
107	37, 106	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
108	107	expcom 413	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝜑 → ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
109	108	a2d 29	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
110	109	adantl 481	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
111	10, 16, 22, 28, 33, 110	findcard2s 9231	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1))))
112	2, 111	mpcom 38	. 2 ⊢ (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1)))
113	1, 112	mpi 20	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ⦋csb 3921   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648   ↦ cmpt 5249  (class class class)co 7448   ∘_f cof 7712  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184   + caddc 11187  𝑂(1)co1 15532  Σcsu 15734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-o1 15536  df-sum 15735

This theorem is referenced by:  rpvmasum2  27574