Theorem fsumo1 15260

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 3899	. 2 ⊢ 𝐵 ⊆ 𝐵
2		fsumo1.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 3902	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		sumeq1 15138	. . . . . . . . 9 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		sum0 15171	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
6	4, 5	eqtrdi 2789	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = 0)
7	6	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
8	7	eleq1d 2817	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1)))
9	3, 8	imbi12d 348	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1)) ↔ (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1))))
10	9	imbi2d 344	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1)))))
11		sseq1 3902	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
12		sumeq1 15138	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝑦 𝐶)
13	12	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
14	13	eleq1d 2817	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1)))
15	11, 14	imbi12d 348	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ∈ 𝑂(1))))
16	15	imbi2d 344	. . . 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 3902	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
18		sumeq1 15138	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
19	18	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
20	19	eleq1d 2817	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))
21	17, 20	imbi12d 348	. . . . 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 344	. . . 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 3902	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
24		sumeq1 15138	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
25	24	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
26	25	eleq1d 2817	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1)))
27	23, 26	imbi12d 348	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝑂(1))))
28	27	imbi2d 344	. . . 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 10711	. . . . . 6 ⊢ 0 ∈ ℂ
31		o1const 15067	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1))
32	29, 30, 31	sylancl 589	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1))
33	32	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ∈ 𝑂(1)))
34		ssun1 4062	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		sstr 3885	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦 ⊆ 𝐵)
36	34, 35	mpan 690	. . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑦)
39		disjsn 4602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
40	38, 39	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
41	40	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∩ {𝑧}) = ∅)
42		eqidd 2739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
43	2	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
44		simprr 773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
45	43, 44	ssfid 8819	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
46	45	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
47	44	sselda 3877	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
48	47	adantlr 715	. . . . . . . . . . . . . . . . . 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 655	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
51		fsumo1.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
52	50, 51	o1mptrcl 15070	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
53	52	an32s 652	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
54	53	adantllr 719	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
55	48, 54	syldan 594	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
56	41, 42, 46, 55	fsumsplit 15190	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
57		nfcv 2899	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑤𝐶
58		nfcsb1v 3814	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐶
59		csbeq1a 3804	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑘⦌𝐶)
60	57, 58, 59	cbvsumi 15147	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶
61	44	unssbd 4078	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
62		vex 3402	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ V
63	62	snss 4674	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
64	61, 63	sylibr 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
65	64	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)
66	54	ralrimiva 3096	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
67		nfcsb1v 3814	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
68	67	nfel1 2915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
69		csbeq1a 3804	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
70	69	eleq1d 2817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
71	68, 70	rspc 3514	. . . . . . . . . . . . . . . . . . . 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 3793	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
74	73	sumsn 15194	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
75	65, 72, 74	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
76	60, 75	syl5eq 2785	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
77	76	oveq2d 7186	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
79	78	mpteq2dva 5125	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
80	29	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ⊆ ℝ)
81		reex 10706	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
82	81	ssex 5189	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
83	80, 82	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ∈ V)
84		sumex 15137	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ 𝑦 𝐶 ∈ V
85	84	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑦 𝐶 ∈ V)
86		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
87		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
88	83, 85, 72, 86, 87	offval2 7444	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 3096	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
93	92	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
94		nfcv 2899	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝐴
95	94, 67	nfmpt 5127	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)
96	95	nfel1 2915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝑂(1)
97	69	mpteq2dv 5126	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
98	97	eleq1d 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝑂(1)))
99	96, 98	rspc 3514	. . . . . . . . . . . . . 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 15061	. . . . . . . . . . . . 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 600	. . . . . . . . . . . 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 2833	. . . . . . . . . . 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 416	. . . . . . . . . 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 460	. . . . . . . . 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 417	. . . . . 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 485	. . . 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 8764	. . 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 399   = wceq 1542   ∈ wcel 2114  ∀wral 3053  Vcvv 3398  ⦋csb 3790   ∪ cun 3841   ∩ cin 3842   ⊆ wss 3843  ∅c0 4211  {csn 4516   ↦ cmpt 5110  (class class class)co 7170   ∘_f cof 7423  Fincfn 8555  ℂcc 10613  ℝcr 10614  0cc0 10615   + caddc 10618  𝑂(1)co1 14933  Σcsu 15135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-ico 12827  df-fz 12982  df-fzo 13125  df-seq 13461  df-exp 13522  df-hash 13783  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-clim 14935  df-rlim 14936  df-o1 14937  df-sum 15136

This theorem is referenced by:  rpvmasum2  26248