sge0resplit - Mathbox for Glauco Siliprandi

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Theorem sge0resplit 46411

Description: Σ^{^} splits into two parts, when it's a real number. This is a special case of sge0split 46414. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
sge0resplit.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sge0resplit.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
sge0resplit.u	⊢ 𝑈 = (𝐴 ∪ 𝐵)
sge0resplit.in0	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
sge0resplit.f	⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))
sge0resplit.re	⊢ (𝜑 → (Σ^{^}‘𝐹) ∈ ℝ)

**Assertion**
Ref	Expression
sge0resplit	⊢ (𝜑 → (Σ^{^}‘𝐹) = ((Σ^{^}‘(𝐹 ↾ 𝐴)) + (Σ^{^}‘(𝐹 ↾ 𝐵))))

Proof of Theorem sge0resplit Dummy variables 𝑎 𝑏 𝑟 𝑢 𝑣 𝑥 𝑦 𝑡 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0resplit.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		sge0resplit.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))
3		ssun1 4144	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
4		sge0resplit.u	. . . . . . . . . . 11 ⊢ 𝑈 = (𝐴 ∪ 𝐵)
5	4	eqcomi 2739	. . . . . . . . . 10 ⊢ (𝐴 ∪ 𝐵) = 𝑈
6	3, 5	sseqtri 3998	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑈
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
8	2, 7	fssresd 6730	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
9	4	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
10		sge0resplit.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
11		unexg 7722	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
12	1, 10, 11	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
13	9, 12	eqeltrd 2829	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ V)
14		sge0resplit.re	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘𝐹) ∈ ℝ)
15	13, 2, 14	sge0ssre 46402	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐴)) ∈ ℝ)
16	1, 8, 15	sge0supre 46394	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ))
17	16, 15	eqeltrrd 2830	. . . . 5 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) ∈ ℝ)
18		ssun2 4145	. . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
19	18, 5	sseqtri 3998	. . . . . . . . 9 ⊢ 𝐵 ⊆ 𝑈
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
21	2, 20	fssresd 6730	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
22	13, 2, 14	sge0ssre 46402	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐵)) ∈ ℝ)
23	10, 21, 22	sge0supre 46394	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < ))
24	23, 22	eqeltrrd 2830	. . . . 5 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < ) ∈ ℝ)
25		rexadd 13199	. . . . 5 ⊢ ((sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) ∈ ℝ ∧ sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < ) ∈ ℝ) → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) +_𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < )) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) + sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < )))
26	17, 24, 25	syl2anc 584	. . . 4 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) +_𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < )) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) + sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < )))
27	13, 2, 14	sge0rern 46393	. . . . . . . 8 ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹)
28		nelrnres 45188	. . . . . . . 8 ⊢ (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
30	8, 29	fge0iccico 46375	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶(0[,)+∞))
31	30	sge0rnre 46369	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ⊆ ℝ)
32		sge0rnn0 46373	. . . . . 6 ⊢ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ≠ ∅
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ≠ ∅)
34	1, 30	sge0reval 46377	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ))
35	34	eqcomd 2736	. . . . . . 7 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ) = (Σ^{^}‘(𝐹 ↾ 𝐴)))
36	35, 15	eqeltrd 2829	. . . . . 6 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ) ∈ ℝ)
37		supxrre3 45328	. . . . . . 7 ⊢ ((ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ⊆ ℝ ∧ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ≠ ∅) → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))𝑡 ≤ 𝑤))
38	31, 33, 37	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))𝑡 ≤ 𝑤))
39	36, 38	mpbid 232	. . . . 5 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))𝑡 ≤ 𝑤)
40		nelrnres 45188	. . . . . . . 8 ⊢ (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
41	27, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
42	21, 41	fge0iccico 46375	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,)+∞))
43	42	sge0rnre 46369	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ⊆ ℝ)
44		sge0rnn0 46373	. . . . . 6 ⊢ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ≠ ∅
45	44	a1i 11	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ≠ ∅)
46	10, 42	sge0reval 46377	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ))
47	46	eqcomd 2736	. . . . . . 7 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ) = (Σ^{^}‘(𝐹 ↾ 𝐵)))
48	47, 22	eqeltrd 2829	. . . . . 6 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ) ∈ ℝ)
49		supxrre3 45328	. . . . . . 7 ⊢ ((ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ⊆ ℝ ∧ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ≠ ∅) → (sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑡 ≤ 𝑤))
50	43, 45, 49	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑡 ≤ 𝑤))
51	48, 50	mpbid 232	. . . . 5 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑡 ≤ 𝑤)
52		eqid 2730	. . . . 5 ⊢ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} = {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}
53	31, 33, 39, 43, 45, 51, 52	supadd 12158	. . . 4 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) + sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < )) = sup({𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}, ℝ, < ))
54		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}) → 𝜑)
55		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
56		eqeq1 2734	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑟 → (𝑧 = (𝑣 + 𝑢) ↔ 𝑟 = (𝑣 + 𝑢)))
57	56	rexbidv 3158	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑟 → (∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢) ↔ ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)))
58	57	rexbidv 3158	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑟 → (∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢) ↔ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)))
59	55, 58	elab 3649	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
60	59	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
61	60	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
62		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)))) → 𝜑)
63		vex 3454	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑣 ∈ V
64		sumeq1 15662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
65	64	cbvmptv 5214	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) = (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
66	65	elrnmpt 5925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ↔ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦)))
67	63, 66	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ↔ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
68	67	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
69	68	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
70		vex 3454	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑢 ∈ V
71		sumeq1 15662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
72	71	cbvmptv 5214	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) = (𝑏 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
73	72	elrnmpt 5925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)))
74	70, 73	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
75	74	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) → ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
76	75	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))) → ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
77	69, 76	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))) → (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)))
78		reeanv 3210	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) ↔ (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)))
79	77, 78	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)))
80	79	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)))
81		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
82		elinel1 4167	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝒫 𝐴)
83		elpwi 4573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
84		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ⊆ 𝐴 → 𝑎 ⊆ 𝐴)
85	84, 6	sstrdi 3962	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ⊆ 𝐴 → 𝑎 ⊆ 𝑈)
86	83, 85	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝑈)
87	82, 86	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ⊆ 𝑈)
88	87	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ⊆ 𝑈)
89		elinel1 4167	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ 𝒫 𝐵)
90		elpwi 4573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
91		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 ⊆ 𝐵 → 𝑏 ⊆ 𝐵)
92	91, 19	sstrdi 3962	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 ⊆ 𝐵 → 𝑏 ⊆ 𝑈)
93	90, 92	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝑈)
94	89, 93	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ⊆ 𝑈)
95	94	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏 ⊆ 𝑈)
96	88, 95	unssd 4158	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎 ∪ 𝑏) ⊆ 𝑈)
97		vex 3454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑎 ∈ V
98		vex 3454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑏 ∈ V
99	97, 98	unex 7723	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∪ 𝑏) ∈ V
100	99	elpw 4570	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∪ 𝑏) ∈ 𝒫 𝑈 ↔ (𝑎 ∪ 𝑏) ⊆ 𝑈)
101	96, 100	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎 ∪ 𝑏) ∈ 𝒫 𝑈)
102		elinel2 4168	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ Fin)
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ Fin)
104		elinel2 4168	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ Fin)
105	104	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏 ∈ Fin)
106		unfi 9141	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ Fin ∧ 𝑏 ∈ Fin) → (𝑎 ∪ 𝑏) ∈ Fin)
107	103, 105, 106	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎 ∪ 𝑏) ∈ Fin)
108	101, 107	elind 4166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎 ∪ 𝑏) ∈ (𝒫 𝑈 ∩ Fin))
109	108	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎 ∪ 𝑏) ∈ (𝒫 𝑈 ∩ Fin))
110	109	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → (𝑎 ∪ 𝑏) ∈ (𝒫 𝑈 ∩ Fin))
111		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) → 𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
112		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) → 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
113	111, 112	oveq12d 7408	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) → (𝑣 + 𝑢) = (Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) + Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)))
114	113	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) → (𝑣 + 𝑢) = (Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) + Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)))
115	82, 83	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ⊆ 𝐴)
116	115	sselda 3949	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑎) → 𝑦 ∈ 𝐴)
117		fvres 6880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
118	116, 117	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑎) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
119	118	sumeq2dv 15675	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑎 (𝐹‘𝑦))
120	119	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑎 (𝐹‘𝑦))
121	89, 90	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ⊆ 𝐵)
122	121	sselda 3949	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑏 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦 ∈ 𝑏) → 𝑦 ∈ 𝐵)
123		fvres 6880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑦) = (𝐹‘𝑦))
124	122, 123	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑏 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦 ∈ 𝑏) → ((𝐹 ↾ 𝐵)‘𝑦) = (𝐹‘𝑦))
125	124	sumeq2dv 15675	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ 𝑏 (𝐹‘𝑦))
126	125	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ 𝑏 (𝐹‘𝑦))
127	120, 126	oveq12d 7408	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) + Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) = (Σ𝑦 ∈ 𝑎 (𝐹‘𝑦) + Σ𝑦 ∈ 𝑏 (𝐹‘𝑦)))
128	127	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) → (Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) + Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) = (Σ𝑦 ∈ 𝑎 (𝐹‘𝑦) + Σ𝑦 ∈ 𝑏 (𝐹‘𝑦)))
129	114, 128	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) → (𝑣 + 𝑢) = (Σ𝑦 ∈ 𝑎 (𝐹‘𝑦) + Σ𝑦 ∈ 𝑏 (𝐹‘𝑦)))
130	129	ad4ant23 753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → (𝑣 + 𝑢) = (Σ𝑦 ∈ 𝑎 (𝐹‘𝑦) + Σ𝑦 ∈ 𝑏 (𝐹‘𝑦)))
131		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 = (𝑣 + 𝑢))
132		sge0resplit.in0	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
133	132	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝐴 ∩ 𝐵) = ∅)
134	115	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → 𝑎 ⊆ 𝐴)
135	121	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏 ⊆ 𝐵)
136	135	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → 𝑏 ⊆ 𝐵)
137		ssin0 45056	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑎 ⊆ 𝐴 ∧ 𝑏 ⊆ 𝐵) → (𝑎 ∩ 𝑏) = ∅)
138	133, 134, 136, 137	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎 ∩ 𝑏) = ∅)
139		eqidd 2731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎 ∪ 𝑏) = (𝑎 ∪ 𝑏))
140	107	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎 ∪ 𝑏) ∈ Fin)
141		rge0ssre 13424	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℝ
142		ax-resscn 11132	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℝ ⊆ ℂ
143	141, 142	sstri 3959	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0[,)+∞) ⊆ ℂ
144	2, 27	fge0iccico 46375	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐹:𝑈⟶(0[,)+∞))
145	144	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → 𝐹:𝑈⟶(0[,)+∞))
146	96	sselda 3949	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → 𝑦 ∈ 𝑈)
147	146	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → 𝑦 ∈ 𝑈)
148	145, 147	ffvelcdmd 7060	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → (𝐹‘𝑦) ∈ (0[,)+∞))
149	143, 148	sselid 3947	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → (𝐹‘𝑦) ∈ ℂ)
150	138, 139, 140, 149	fsumsplit 15714	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → Σ𝑦 ∈ (𝑎 ∪ 𝑏)(𝐹‘𝑦) = (Σ𝑦 ∈ 𝑎 (𝐹‘𝑦) + Σ𝑦 ∈ 𝑏 (𝐹‘𝑦)))
151	150	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → Σ𝑦 ∈ (𝑎 ∪ 𝑏)(𝐹‘𝑦) = (Σ𝑦 ∈ 𝑎 (𝐹‘𝑦) + Σ𝑦 ∈ 𝑏 (𝐹‘𝑦)))
152	130, 131, 151	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 = Σ𝑦 ∈ (𝑎 ∪ 𝑏)(𝐹‘𝑦))
153		sumeq1 15662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑎 ∪ 𝑏) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = Σ𝑦 ∈ (𝑎 ∪ 𝑏)(𝐹‘𝑦))
154	153	rspceeqv 3614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑎 ∪ 𝑏) ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ (𝑎 ∪ 𝑏)(𝐹‘𝑦)) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
155	110, 152, 154	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
156	55	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 ∈ V)
157	81, 155, 156	elrnmptd 5930	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
158	157	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
159	158	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → ((𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))))
160	159	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → ((𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))))
161	160	rexlimdvv 3194	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦)) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))))
162	161	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
163	62, 80, 162	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
164	163	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))))
165	164	rexlimdvv 3194	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
166	165	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
167	54, 61, 166	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
168	167	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
169	81	elrnmpt 5925	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
170	169	ibi 267	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
171	170	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
172		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝜑
173		nfcv 2892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑟
174		nfmpt1 5209	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
175	174	nfrn 5919	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
176	173, 175	nfel 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
177	172, 176	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
178		nfmpt1 5209	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))
179	178	nfrn 5919	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))
180		nfmpt1 5209	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))
181	180	nfrn 5919	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))
182		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝑟 = (𝑣 + 𝑢)
183	181, 182	nfrexw 3289	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)
184	179, 183	nfrexw 3289	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)
185		inss2 4204	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
186	185	sseli 3945	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐴) → 𝑦 ∈ 𝐴)
187	186	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → 𝑦 ∈ 𝐴)
188	117	eqcomd 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = ((𝐹 ↾ 𝐴)‘𝑦))
189	187, 188	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → (𝐹‘𝑦) = ((𝐹 ↾ 𝐴)‘𝑦))
190	189	sumeq2dv 15675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦))
191		sumeq1 15662	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦))
192	191	cbvmptv 5214	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) = (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦))
193		vex 3454	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑥 ∈ V
194	193	inex1 5275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∩ 𝐴) ∈ V
195	194	elpw 4570	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴)
196	185, 195	mpbir 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴
197	196	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴)
198		elinel2 4168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
199		inss1 4203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
200	199	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ⊆ 𝑥)
201		ssfi 9143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ Fin ∧ (𝑥 ∩ 𝐴) ⊆ 𝑥) → (𝑥 ∩ 𝐴) ∈ Fin)
202	198, 200, 201	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ∈ Fin)
203	197, 202	elind 4166	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ∈ (𝒫 𝐴 ∩ Fin))
204		eqidd 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦))
205		sumeq1 15662	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑥 ∩ 𝐴) → Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦))
206	205	rspceeqv 3614	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∩ 𝐴) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦)) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦))
207	203, 204, 206	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦))
208		sumex 15661	. . . . . . . . . . . . . . . . . . 19 ⊢ Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) ∈ V
209	208	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) ∈ V)
210	192, 207, 209	elrnmptd 5930	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)))
211	190, 210	eqeltrd 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)))
212	211	3ad2ant2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)))
213		sumeq1 15662	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦))
214	213	cbvmptv 5214	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) = (𝑧 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦))
215		inss2 4204	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
216	193	inex1 5275	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ 𝐵) ∈ V
217	216	elpw 4570	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∩ 𝐵) ∈ 𝒫 𝐵 ↔ (𝑥 ∩ 𝐵) ⊆ 𝐵)
218	215, 217	mpbir 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∩ 𝐵) ∈ 𝒫 𝐵
219	218	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → (𝑥 ∩ 𝐵) ∈ 𝒫 𝐵)
220		inss1 4203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝑥
221	220	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐵) ⊆ 𝑥)
222		ssfi 9143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ Fin ∧ (𝑥 ∩ 𝐵) ⊆ 𝑥) → (𝑥 ∩ 𝐵) ∈ Fin)
223	198, 221, 222	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐵) ∈ Fin)
224	223	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → (𝑥 ∩ 𝐵) ∈ Fin)
225	219, 224	elind 4166	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → (𝑥 ∩ 𝐵) ∈ (𝒫 𝐵 ∩ Fin))
226	215	sseli 3945	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → 𝑦 ∈ 𝐵)
227	123	eqcomd 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐵 → (𝐹‘𝑦) = ((𝐹 ↾ 𝐵)‘𝑦))
228	226, 227	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → (𝐹‘𝑦) = ((𝐹 ↾ 𝐵)‘𝑦))
229	228	sumeq2i 15671	. . . . . . . . . . . . . . . . . . . 20 ⊢ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦)
230	229	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦))
231	230	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦))
232		sumeq1 15662	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦))
233	232	rspceeqv 3614	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∩ 𝐵) ∈ (𝒫 𝐵 ∩ Fin) ∧ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦)) → ∃𝑧 ∈ (𝒫 𝐵 ∩ Fin)Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦))
234	225, 231, 233	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → ∃𝑧 ∈ (𝒫 𝐵 ∩ Fin)Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦))
235		sumex 15661	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ V
236	235	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ V)
237	214, 234, 236	elrnmptd 5930	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)))
238		simp3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
239	185	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑥 ∩ 𝐴) ⊆ 𝐴)
240	215	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑥 ∩ 𝐵) ⊆ 𝐵)
241		ssin0 45056	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ (𝑥 ∩ 𝐵) ⊆ 𝐵) → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
242	132, 239, 240, 241	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
243	242	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
244		elinel1 4167	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
245		elpwi 4573	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝒫 𝑈 → 𝑥 ⊆ 𝑈)
246	244, 245	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ⊆ 𝑈)
247	4	ineq2i 4183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∩ 𝑈) = (𝑥 ∩ (𝐴 ∪ 𝐵))
248	247	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑈 → (𝑥 ∩ 𝑈) = (𝑥 ∩ (𝐴 ∪ 𝐵)))
249		dfss 3936	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ⊆ 𝑈 ↔ 𝑥 = (𝑥 ∩ 𝑈))
250	249	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑈 → 𝑥 = (𝑥 ∩ 𝑈))
251		indi 4250	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵))
252	251	eqcomi 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)) = (𝑥 ∩ (𝐴 ∪ 𝐵))
253	252	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑈 → ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)) = (𝑥 ∩ (𝐴 ∪ 𝐵)))
254	248, 250, 253	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ⊆ 𝑈 → 𝑥 = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)))
255	246, 254	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)))
256	255	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)))
257	198	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
258	144	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑈⟶(0[,)+∞))
259	246	sselda 3949	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
260	259	adantll 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
261	258, 260	ffvelcdmd 7060	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
262	143, 261	sselid 3947	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
263	243, 256, 257, 262	fsumsplit 15714	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
264	263	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
265	238, 264	eqtrd 2765	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → 𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
266		oveq2 7398	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
267	266	rspceeqv 3614	. . . . . . . . . . . . . . . 16 ⊢ ((Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ∧ 𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦))) → ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢))
268	237, 265, 267	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢))
269		oveq1 7397	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) → (𝑣 + 𝑢) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢))
270	269	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) → (𝑟 = (𝑣 + 𝑢) ↔ 𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢)))
271	270	rexbidv 3158	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) → (∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢) ↔ ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢)))
272	271	rspcev 3591	. . . . . . . . . . . . . . 15 ⊢ ((Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ∧ ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢)) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
273	212, 268, 272	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
274	273	3exp 1119	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))))
275	274	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))))
276	177, 184, 275	rexlimd 3245	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → (∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)))
277	171, 276	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
278	277, 59	sylibr 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → 𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)})
279	278	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → 𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}))
280	168, 279	impbid 212	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
281	280	alrimiv 1927	. . . . . 6 ⊢ (𝜑 → ∀𝑟(𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
282		dfcleq 2723	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} = ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∀𝑟(𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))))
283	281, 282	sylibr 234	. . . . 5 ⊢ (𝜑 → {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} = ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
284	283	supeq1d 9404	. . . 4 ⊢ (𝜑 → sup({𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}, ℝ, < ) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ, < ))
285	26, 53, 284	3eqtrrd 2770	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ, < ) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) +_𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < )))
286	13, 2, 14	sge0supre 46394	. . 3 ⊢ (𝜑 → (Σ^{^}‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ, < ))
287	16, 23	oveq12d 7408	. . 3 ⊢ (𝜑 → ((Σ^{^}‘(𝐹 ↾ 𝐴)) +_𝑒 (Σ^{^}‘(𝐹 ↾ 𝐵))) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) +_𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < )))
288	285, 286, 287	3eqtr4d 2775	. 2 ⊢ (𝜑 → (Σ^{^}‘𝐹) = ((Σ^{^}‘(𝐹 ↾ 𝐴)) +_𝑒 (Σ^{^}‘(𝐹 ↾ 𝐵))))
289		rexadd 13199	. . 3 ⊢ (((Σ^{^}‘(𝐹 ↾ 𝐴)) ∈ ℝ ∧ (Σ^{^}‘(𝐹 ↾ 𝐵)) ∈ ℝ) → ((Σ^{^}‘(𝐹 ↾ 𝐴)) +_𝑒 (Σ^{^}‘(𝐹 ↾ 𝐵))) = ((Σ^{^}‘(𝐹 ↾ 𝐴)) + (Σ^{^}‘(𝐹 ↾ 𝐵))))
290	15, 22, 289	syl2anc 584	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝐹 ↾ 𝐴)) +_𝑒 (Σ^{^}‘(𝐹 ↾ 𝐵))) = ((Σ^{^}‘(𝐹 ↾ 𝐴)) + (Σ^{^}‘(𝐹 ↾ 𝐵))))
291	288, 290	eqtrd 2765	1 ⊢ (𝜑 → (Σ^{^}‘𝐹) = ((Σ^{^}‘(𝐹 ↾ 𝐴)) + (Σ^{^}‘(𝐹 ↾ 𝐵))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∀wal 1538 = wceq 1540 ∈ wcel 2109 {cab 2708 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ∪ cun 3915 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 𝒫 cpw 4566 class class class wbr 5110 ↦ cmpt 5191 ran crn 5642 ↾ cres 5643 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 supcsup 9398 ℂcc 11073 ℝcr 11074 0cc0 11075 + caddc 11078 +∞cpnf 11212 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 +_𝑒 cxad 13077 [,)cico 13315 [,]cicc 13316 Σcsu 15659 Σ^{^}csumge0 46367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-xadd 13080 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-sumge0 46368

This theorem is referenced by: sge0split 46414

W3C validator