sge0resplit - Mathbox for Glauco Siliprandi

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Theorem sge0resplit 46852

Description: Σ^{^} splits into two parts, when it's a real number. This is a special case of sge0split 46855. (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 4119	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
4		sge0resplit.u	. . . . . . . . . . 11 ⊢ 𝑈 = (𝐴 ∪ 𝐵)
5	4	eqcomi 2746	. . . . . . . . . 10 ⊢ (𝐴 ∪ 𝐵) = 𝑈
6	3, 5	sseqtri 3971	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑈
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
8	2, 7	fssresd 6701	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
9	4	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
10		sge0resplit.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
11		unexg 7690	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
12	1, 10, 11	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
13	9, 12	eqeltrd 2837	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ V)
14		sge0resplit.re	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘𝐹) ∈ ℝ)
15	13, 2, 14	sge0ssre 46843	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐴)) ∈ ℝ)
16	1, 8, 15	sge0supre 46835	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ))
17	16, 15	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ, < ) ∈ ℝ)
18		ssun2 4120	. . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
19	18, 5	sseqtri 3971	. . . . . . . . 9 ⊢ 𝐵 ⊆ 𝑈
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
21	2, 20	fssresd 6701	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
22	13, 2, 14	sge0ssre 46843	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐵)) ∈ ℝ)
23	10, 21, 22	sge0supre 46835	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < ))
24	23, 22	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ, < ) ∈ ℝ)
25		rexadd 13175	. . . . 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 585	. . . 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 46834	. . . . . . . 8 ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹)
28		nelrnres 45635	. . . . . . . 8 ⊢ (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
30	8, 29	fge0iccico 46816	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶(0[,)+∞))
31	30	sge0rnre 46810	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) ⊆ ℝ)
32		sge0rnn0 46814	. . . . . 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 46818	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ))
35	34	eqcomd 2743	. . . . . . 7 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ) = (Σ^{^}‘(𝐹 ↾ 𝐴)))
36	35, 15	eqeltrd 2837	. . . . . 6 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)), ℝ^*, < ) ∈ ℝ)
37		supxrre3 45773	. . . . . . 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 585	. . . . . 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 45635	. . . . . . . 8 ⊢ (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
41	27, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
42	21, 41	fge0iccico 46816	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,)+∞))
43	42	sge0rnre 46810	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) ⊆ ℝ)
44		sge0rnn0 46814	. . . . . 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 46818	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝐹 ↾ 𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ))
47	46	eqcomd 2743	. . . . . . 7 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ) = (Σ^{^}‘(𝐹 ↾ 𝐵)))
48	47, 22	eqeltrd 2837	. . . . . 6 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)), ℝ^*, < ) ∈ ℝ)
49		supxrre3 45773	. . . . . . 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 585	. . . . . 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 2737	. . . . 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 12115	. . . 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 3434	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
56		eqeq1 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑟 → (𝑧 = (𝑣 + 𝑢) ↔ 𝑟 = (𝑣 + 𝑢)))
57	56	rexbidv 3162	. . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . 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 3623	. . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑣 ∈ V
64		sumeq1 15642	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
65	64	cbvmptv 5190	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) = (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑦))
66	65	elrnmpt 5907	. . . . . . . . . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑢 ∈ V
71		sumeq1 15642	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
72	71	cbvmptv 5190	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) = (𝑏 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑏 ((𝐹 ↾ 𝐵)‘𝑦))
73	72	elrnmpt 5907	. . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
82		elinel1 4142	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝒫 𝐴)
83		elpwi 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
84		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ⊆ 𝐴 → 𝑎 ⊆ 𝐴)
85	84, 6	sstrdi 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ 𝒫 𝐵)
90		elpwi 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 ∈ 𝒫 𝐵 → 𝑏 ⊆ 𝐵)
91		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 ⊆ 𝐵 → 𝑏 ⊆ 𝐵)
92	91, 19	sstrdi 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎 ∪ 𝑏) ⊆ 𝑈)
97		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑎 ∈ V
98		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑏 ∈ V
99	97, 98	unex 7691	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∪ 𝑏) ∈ V
100	99	elpw 4546	. . . . . . . . . . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ Fin)
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ Fin)
104		elinel2 4143	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ Fin)
105	104	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏 ∈ Fin)
106		unfi 9098	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ Fin ∧ 𝑏 ∈ Fin) → (𝑎 ∪ 𝑏) ∈ Fin)
107	103, 105, 106	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎 ∪ 𝑏) ∈ Fin)
108	101, 107	elind 4141	. . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑎) → 𝑦 ∈ 𝐴)
117		fvres 6853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15655	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑏 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦 ∈ 𝑏) → 𝑦 ∈ 𝐵)
123		fvres 6853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15655	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . . . 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 754	. . . . . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . . . . . . . . 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 45504	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑎 ⊆ 𝐴 ∧ 𝑏 ⊆ 𝐵) → (𝑎 ∩ 𝑏) = ∅)
138	133, 134, 136, 137	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎 ∩ 𝑏) = ∅)
139		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . 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 13400	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℝ
142		ax-resscn 11086	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℝ ⊆ ℂ
143	141, 142	sstri 3932	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0[,)+∞) ⊆ ℂ
144	2, 27	fge0iccico 46816	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐹:𝑈⟶(0[,)+∞))
145	144	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → 𝐹:𝑈⟶(0[,)+∞))
146	96	sselda 3922	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → 𝑦 ∈ 𝑈)
147	146	adantll 715	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → 𝑦 ∈ 𝑈)
148	145, 147	ffvelcdmd 7031	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → (𝐹‘𝑦) ∈ (0[,)+∞))
149	143, 148	sselid 3920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎 ∪ 𝑏)) → (𝐹‘𝑦) ∈ ℂ)
150	138, 139, 140, 149	fsumsplit 15694	. . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . . . . . 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 15642	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑎 ∪ 𝑏) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = Σ𝑦 ∈ (𝑎 ∪ 𝑏)(𝐹‘𝑦))
154	153	rspceeqv 3588	. . . . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . . . 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 5912	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 585	. . . . . . . . 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 5907	. . . . . . . . . . . . 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 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝜑
173		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑟
174		nfmpt1 5185	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
175	174	nfrn 5901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
176	173, 175	nfel 2914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
177	172, 176	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
178		nfmpt1 5185	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))
179	178	nfrn 5901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦))
180		nfmpt1 5185	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))
181	180	nfrn 5901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))
182		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝑟 = (𝑣 + 𝑢)
183	181, 182	nfrexw 3286	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)
184	179, 183	nfrexw 3286	. . . . . . . . . . . 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 4179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
186	185	sseli 3918	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐴) → 𝑦 ∈ 𝐴)
187	186	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → 𝑦 ∈ 𝐴)
188	117	eqcomd 2743	. . . . . . . . . . . . . . . . . . 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 15655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦))
191		sumeq1 15642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦))
192	191	cbvmptv 5190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)) = (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦))
193		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑥 ∈ V
194	193	inex1 5254	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∩ 𝐴) ∈ V
195	194	elpw 4546	. . . . . . . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
199		inss1 4178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
200	199	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ⊆ 𝑥)
201		ssfi 9100	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ Fin ∧ (𝑥 ∩ 𝐴) ⊆ 𝑥) → (𝑥 ∩ 𝐴) ∈ Fin)
202	198, 200, 201	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ∈ Fin)
203	197, 202	elind 4141	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ∈ (𝒫 𝐴 ∩ Fin))
204		eqidd 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦))
205		sumeq1 15642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑥 ∩ 𝐴) → Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦))
206	205	rspceeqv 3588	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐴)‘𝑦))
208		sumex 15641	. . . . . . . . . . . . . . . . . . 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 5912	. . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐴)‘𝑦)))
212	211	3ad2ant2 1135	. . . . . . . . . . . . . . 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 15642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦))
214	213	cbvmptv 5190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝐵)‘𝑦)) = (𝑧 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦))
215		inss2 4179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
216	193	inex1 5254	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ 𝐵) ∈ V
217	216	elpw 4546	. . . . . . . . . . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝑥
221	220	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐵) ⊆ 𝑥)
222		ssfi 9100	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ Fin ∧ (𝑥 ∩ 𝐵) ⊆ 𝑥) → (𝑥 ∩ 𝐵) ∈ Fin)
223	198, 221, 222	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐵) ∈ Fin)
224	223	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → (𝑥 ∩ 𝐵) ∈ Fin)
225	219, 224	elind 4141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → (𝑥 ∩ 𝐵) ∈ (𝒫 𝐵 ∩ Fin))
226	215	sseli 3918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → 𝑦 ∈ 𝐵)
227	123	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . 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 15651	. . . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦))
232		sumeq1 15642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → Σ𝑦 ∈ 𝑧 ((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦))
233	232	rspceeqv 3588	. . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . 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 15641	. . . . . . . . . . . . . . . . . 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 5912	. . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . 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 45504	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ (𝑥 ∩ 𝐵) ⊆ 𝐵) → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
242	132, 239, 240, 241	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
243	242	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
244		elinel1 4142	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
245		elpwi 4549	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝒫 𝑈 → 𝑥 ⊆ 𝑈)
246	244, 245	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ⊆ 𝑈)
247	4	ineq2i 4158	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∩ 𝑈) = (𝑥 ∩ (𝐴 ∪ 𝐵))
248	247	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑈 → (𝑥 ∩ 𝑈) = (𝑥 ∩ (𝐴 ∪ 𝐵)))
249		dfss 3909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ⊆ 𝑈 ↔ 𝑥 = (𝑥 ∩ 𝑈))
250	249	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑈 → 𝑥 = (𝑥 ∩ 𝑈))
251		indi 4225	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵))
252	251	eqcomi 2746	. . . . . . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑈⟶(0[,)+∞))
259	246	sselda 3922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
260	259	adantll 715	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
261	258, 260	ffvelcdmd 7031	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
262	143, 261	sselid 3920	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
263	243, 256, 257, 262	fsumsplit 15694	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
264	263	3adant3 1133	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → 𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
266		oveq2 7368	. . . . . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 7367	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) → (𝑣 + 𝑢) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢))
270	269	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) → (𝑟 = (𝑣 + 𝑢) ↔ 𝑟 = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + 𝑢)))
271	270	rexbidv 3162	. . . . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . 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 1929	. . . . . 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 2730	. . . . . 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 9352	. . . 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 2777	. . 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 46835	. . 3 ⊢ (𝜑 → (Σ^{^}‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ, < ))
287	16, 23	oveq12d 7378	. . 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 2782	. 2 ⊢ (𝜑 → (Σ^{^}‘𝐹) = ((Σ^{^}‘(𝐹 ↾ 𝐴)) +_𝑒 (Σ^{^}‘(𝐹 ↾ 𝐵))))
289		rexadd 13175	. . 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 585	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝐹 ↾ 𝐴)) +_𝑒 (Σ^{^}‘(𝐹 ↾ 𝐵))) = ((Σ^{^}‘(𝐹 ↾ 𝐴)) + (Σ^{^}‘(𝐹 ↾ 𝐵))))
291	288, 290	eqtrd 2772	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 1087 ∀wal 1540 = wceq 1542 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 𝒫 cpw 4542 class class class wbr 5086 ↦ cmpt 5167 ran crn 5625 ↾ cres 5626 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 Fincfn 8886 supcsup 9346 ℂcc 11027 ℝcr 11028 0cc0 11029 + caddc 11032 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 +_𝑒 cxad 13052 [,)cico 13291 [,]cicc 13292 Σcsu 15639 Σ^{^}csumge0 46808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-xadd 13055 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-sumge0 46809

This theorem is referenced by: sge0split 46855

W3C validator