esum2d - Mathbox for Thierry Arnoux

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Theorem esum2d 32061

Description: Write a double extended sum as a sum over a two-dimensional region. Note that 𝐵(𝑗) is a function of 𝑗. This can be seen as "slicing" the relation 𝐴. (Contributed by Thierry Arnoux, 17-May-2020.)

**Hypotheses**
Ref	Expression
esum2d.0	⊢ Ⅎ𝑘𝐹
esum2d.1	⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)
esum2d.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esum2d.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
esum2d.4	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))

**Assertion**
Ref	Expression
esum2d	⊢ (𝜑 → Σ^𝑗 ∈ 𝐴Σ^𝑘 ∈ 𝐵𝐶 = Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)

Distinct variable groups: 𝑗,𝑘,𝐴,𝑧 𝑧,𝐶 𝐵,𝑘,𝑧 𝑗,𝐹 𝑗,𝑊,𝑘 𝜑,𝑗,𝑘,𝑧 Allowed substitution hints: 𝐵(𝑗) 𝐶(𝑗,𝑘) 𝐹(𝑧,𝑘) 𝑉(𝑧,𝑗,𝑘) 𝑊(𝑧)

Proof of Theorem esum2d Dummy variables 𝑡 𝑎 𝑐 𝑟 𝑠 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 12875	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑐𝜑
4		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑐𝑠
5		nfmpt1 5182	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
6	5	nfrn 5861	. . . . . . . . . 10 ⊢ Ⅎ𝑐ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
7	4, 6	nfel 2921	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
8	3, 7	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑐(𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
9		iccssxr 13162	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ^*
10		xrge0base 31294	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11		xrge0cmn 20640	. . . . . . . . . . . . . . . 16 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
12	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
13		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
14	13	elin2d 4133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ Fin)
15		simpll 764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝜑)
16	13	elin1d 4132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
17	16	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
18		vex 3436	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑐 ∈ V
19	18	elpw 4537	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
20	17, 19	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
21		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ 𝑐)
22	20, 21	sseldd 3922	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
23		nfv 1917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗𝜑
24		nfcv 2907	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗𝑧
25		nfiu1 4958	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
26	24, 25	nfel 2921	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
27	23, 26	nfan 1902	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
28		nfv 1917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵))
29		esum2d.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘𝐹
30		nfcv 2907	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(0[,]+∞)
31	29, 30	nfel 2921	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘 𝐹 ∈ (0[,]+∞)
32		esum2d.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)
33	32	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 = 𝐶)
34		simp-5l 782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝜑)
35		simp-4r 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝑗 ∈ 𝐴)
36		simplr 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝑘 ∈ 𝐵)
37		esum2d.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
38	34, 35, 36, 37	syl12anc 834	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐶 ∈ (0[,]+∞))
39	33, 38	eqeltrd 2839	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 ∈ (0[,]+∞))
40		elsnxp 6194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝐴 → (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩))
41	40	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
42	41	adantll 711	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
43	28, 31, 39, 42	r19.29af2 3261	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
44		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
45		eliun 4928	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
46	44, 45	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
47	27, 43, 46	r19.29af 3262	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
48	15, 22, 47	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝐹 ∈ (0[,]+∞))
49	48	ralrimiva 3103	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∀𝑧 ∈ 𝑐 𝐹 ∈ (0[,]+∞))
50	10, 12, 14, 49	gsummptcl 19568	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ (0[,]+∞))
51	9, 50	sselid 3919	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
52	51	ralrimiva 3103	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
53		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) = (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
54	53	rnmptss 6996	. . . . . . . . . . . 12 ⊢ (∀𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^ → ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ⊆ ℝ^)
55	52, 54	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ⊆ ℝ^)
56	55	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ⊆ ℝ^)
57		simpllr 773	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
58	56, 57	sseldd 3922	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 ∈ ℝ^*)
59		esum2d.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
60		snex 5354	. . . . . . . . . . . . . . 15 ⊢ {𝑗} ∈ V
61		esum2d.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
62		xpexg 7600	. . . . . . . . . . . . . . 15 ⊢ (({𝑗} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝑗} × 𝐵) ∈ V)
63	60, 61, 62	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
64	63	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
65		iunexg 7806	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
66	59, 64, 65	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
67	47	ralrimiva 3103	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
68		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
69	68	esumcl 31998	. . . . . . . . . . . 12 ⊢ ((∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V ∧ ∀𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞)) → Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
70	66, 67, 69	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
71	9, 70	sselid 3919	. . . . . . . . . 10 ⊢ (𝜑 → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
72	71	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
73		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
74		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
75		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝑐
76	74, 75, 14, 48	esumgsum 32013	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
77	66	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
78	47	adantlr 712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
79	16, 19	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
80	74, 77, 78, 79	esummono 32022	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
81	76, 80	eqbrtrrd 5098	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
82	81	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ≤ Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
83	82	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
84	73, 83	eqbrtrd 5096	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 ≤ Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
85		xrlenlt 11040	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℝ^* ∧ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^) → (𝑠 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ↔ ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
86	85	biimpa 477	. . . . . . . . 9 ⊢ (((𝑠 ∈ ℝ^* ∧ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^) ∧ 𝑠 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠)
87	58, 72, 84, 86	syl21anc 835	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → ¬ Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠)
88		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) → 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
89		ovex 7308	. . . . . . . . . 10 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ V
90	53, 89	elrnmpti 5869	. . . . . . . . 9 ⊢ (𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ↔ ∃𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
91	88, 90	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) → ∃𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
92	8, 87, 91	r19.29af 3262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) → ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠)
93	92	ralrimiva 3103	. . . . . 6 ⊢ (𝜑 → ∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠)
94		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
95		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑐 𝑠 < 𝑡
96	6, 95	nfrex 3242	. . . . . . . . 9 ⊢ Ⅎ𝑐∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡
97	76	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
98	97	adantlr 712	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
99	98	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
100		simplr 766	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^*𝑧 ∈ 𝑐𝐹) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
101	89	a1i 11	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ V)
102	53	elrnmpt1 5867	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ∧ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ V) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
103	100, 101, 102	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
104	99, 103	eqeltrd 2839	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → Σ^𝑧 ∈ 𝑐𝐹 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
105		simpr 485	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) ∧ 𝑡 = Σ^𝑧 ∈ 𝑐𝐹) → 𝑡 = Σ^*𝑧 ∈ 𝑐𝐹)
106	105	breq2d 5086	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) ∧ 𝑡 = Σ^𝑧 ∈ 𝑐𝐹) → (𝑠 < 𝑡 ↔ 𝑠 < Σ^*𝑧 ∈ 𝑐𝐹))
107		simpr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → 𝑠 < Σ^𝑧 ∈ 𝑐𝐹)
108	104, 106, 107	rspcedvd 3563	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)
109		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑠 ∈ ℝ^*)
110		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝑠
111		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧 <
112	68	nfesum1 32008	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
113	110, 111, 112	nfbr 5121	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
114	109, 113	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
115	66	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
116	47	3ad2antr3 1189	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → 𝐹 ∈ (0[,]+∞))
117	116	3anassrs 1359	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
118		simplr 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑠 ∈ ℝ^*)
119		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
120	114, 115, 117, 118, 119	esumlub 32028	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ∃𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)𝑠 < Σ^*𝑧 ∈ 𝑐𝐹)
121	94, 96, 108, 120	r19.29af2 3261	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)
122	121	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ^) → (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡))
123	122	ralrimiva 3103	. . . . . 6 ⊢ (𝜑 → ∀𝑠 ∈ ℝ^* (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡))
124	93, 123	jca 512	. . . . 5 ⊢ (𝜑 → (∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠 ∧ ∀𝑠 ∈ ℝ^* (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)))
125		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
126	125	breq1d 5084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (𝑟 < 𝑠 ↔ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
127	126	notbid 318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (¬ 𝑟 < 𝑠 ↔ ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
128	127	ralbidv 3112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ 𝑟 < 𝑠 ↔ ∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
129	125	breq2d 5086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (𝑠 < 𝑟 ↔ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹))
130	129	imbi1d 342	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ((𝑠 < 𝑟 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡) ↔ (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)))
131	130	ralbidv 3112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (∀𝑠 ∈ ℝ^ (𝑠 < 𝑟 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡) ↔ ∀𝑠 ∈ ℝ^ (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)))
132	128, 131	anbi12d 631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ((∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ 𝑟 < 𝑠 ∧ ∀𝑠 ∈ ℝ^* (𝑠 < 𝑟 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)) ↔ (∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠 ∧ ∀𝑠 ∈ ℝ^ (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡))))
133	71, 132	rspcedv 3554	. . . . 5 ⊢ (𝜑 → ((∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠 ∧ ∀𝑠 ∈ ℝ^* (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)) → ∃𝑟 ∈ ℝ^* (∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ 𝑟 < 𝑠 ∧ ∀𝑠 ∈ ℝ^ (𝑠 < 𝑟 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡))))
134	124, 133	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑟 ∈ ℝ^* (∀𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ¬ 𝑟 < 𝑠 ∧ ∀𝑠 ∈ ℝ^ (𝑠 < 𝑟 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)))
135	2, 134	supcl 9217	. . 3 ⊢ (𝜑 → sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) ∈ ℝ^*)
136		nfv 1917	. . . . 5 ⊢ Ⅎ𝑎𝜑
137		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑎𝑠
138		nfmpt1 5182	. . . . . . 7 ⊢ Ⅎ𝑎(𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
139	138	nfrn 5861	. . . . . 6 ⊢ Ⅎ𝑎ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
140	137, 139	nfel 2921	. . . . 5 ⊢ Ⅎ𝑎 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
141	136, 140	nfan 1902	. . . 4 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
142		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
143		simpll 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝜑)
144	142	elin1d 4132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ 𝒫 𝐴)
145		elpwi 4542	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
146	144, 145	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ⊆ 𝐴)
147	146	sselda 3921	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝑗 ∈ 𝐴)
148	143, 147, 61	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝐵 ∈ 𝑊)
149	143	adantrr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝜑)
150	147	adantrr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑗 ∈ 𝐴)
151		simprr 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑘 ∈ 𝐵)
152	149, 150, 151, 37	syl12anc 834	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
153	142	elin2d 4133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ Fin)
154	29, 32, 142, 148, 152, 153	esum2dlem 32060	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑗 ∈ 𝑎Σ^𝑘 ∈ 𝐵𝐶 = Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹)
155		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
156		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑎
157	37	anassrs 468	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
158	157	ralrimiva 3103	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
159		nfcv 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐵
160	159	esumcl 31998	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
161	61, 158, 160	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
162	143, 147, 161	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
163	155, 156, 153, 162	esumgsum 32013	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑗 ∈ 𝑎Σ^𝑘 ∈ 𝐵𝐶 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
164	154, 163	eqtr3d 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^*𝑘 ∈ 𝐵𝐶)))
165		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
166	66	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
167	47	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
168		iunss1 4938	. . . . . . . . . . . 12 ⊢ (𝑎 ⊆ 𝐴 → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
169	146, 168	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
170	165, 166, 167, 169	esummono 32022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
171	164, 170	eqbrtrrd 5098	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ≤ Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
172	11	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
173	162	ralrimiva 3103	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑗 ∈ 𝑎 Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
174	10, 172, 153, 173	gsummptcl 19568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ (0[,]+∞))
175	9, 174	sselid 3919	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ ℝ^*)
176	71	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
177		xrlenlt 11040	. . . . . . . . . 10 ⊢ ((((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ ℝ^* ∧ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^) → (((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ↔ ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
178	175, 176, 177	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → (((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ↔ ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
179	171, 178	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^*𝑘 ∈ 𝐵𝐶)))
180		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝜑
181		eqidd 2739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
182	180, 68, 66, 47, 181	esumval 32014	. . . . . . . . . 10 ⊢ (𝜑 → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 = sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^*, < ))
183	182	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 = sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^*, < ))
184	183	breq1d 5084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ↔ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^*𝑘 ∈ 𝐵𝐶))))
185	179, 184	mtbid 324	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
186	185	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
187	186	adantr 481	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → ¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
188		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
189	188	breq2d 5086	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → (sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < 𝑠 ↔ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
190	189	notbid 318	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → (¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < 𝑠 ↔ ¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
191	187, 190	mpbird 256	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → ¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < 𝑠)
192		eqid 2738	. . . . . . 7 ⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) = (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
193		ovex 7308	. . . . . . 7 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V
194	192, 193	elrnmpti 5869	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) ↔ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
195	194	biimpi 215	. . . . 5 ⊢ (𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
196	195	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
197	141, 191, 196	r19.29af 3262	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) → ¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < 𝑠)
198	4	nfel1 2923	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ℝ^*
199		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑐 <
200		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑐ℝ^*
201	6, 200, 199	nfsup 9210	. . . . . . . . . 10 ⊢ Ⅎ𝑐sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )
202	4, 199, 201	nfbr 5121	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )
203	198, 202	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑐(𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))
204	3, 203	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑐(𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )))
205		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑐𝑢
206	205, 6	nfel 2921	. . . . . . 7 ⊢ Ⅎ𝑐 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
207	204, 206	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑐((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
208		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑐 𝑠 < 𝑢
209	207, 208	nfan 1902	. . . . 5 ⊢ Ⅎ𝑐(((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢)
210		simp-5l 782	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝜑)
211		simpr1l 1229	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ (𝑠 < 𝑢 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))) → 𝑠 ∈ ℝ^*)
212	211	3anassrs 1359	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ (𝑠 < 𝑢 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) → 𝑠 ∈ ℝ^*)
213	212	3anassrs 1359	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 ∈ ℝ^*)
214	210, 213	jca 512	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → (𝜑 ∧ 𝑠 ∈ ℝ^*))
215		simpr1r 1230	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ (𝑠 < 𝑢 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))) → 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))
216	215	3anassrs 1359	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ (𝑠 < 𝑢 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) → 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))
217	216	3anassrs 1359	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))
218	214, 217	jca 512	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → ((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^*, < )))
219		simpllr 773	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 < 𝑢)
220		simpr 485	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑢 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
221	219, 220	breqtrd 5100	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 < ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
222		simplr 766	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
223		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
224	223	elin1d 4132	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
225		elpwi 4542	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
226		dmss 5811	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
227		dmiun 5822	. . . . . . . . . . . . . 14 ⊢ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵)
228	226, 227	sseqtrdi 3971	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵))
229		dmxpss 6074	. . . . . . . . . . . . . . . . 17 ⊢ dom ({𝑗} × 𝐵) ⊆ {𝑗}
230	229	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ {𝑗})
231		snssi 4741	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → {𝑗} ⊆ 𝐴)
232	230, 231	sstrd 3931	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ 𝐴)
233	232	rgen 3074	. . . . . . . . . . . . . 14 ⊢ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
234		iunss 4975	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴 ↔ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴)
235	233, 234	mpbir 230	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
236	228, 235	sstrdi 3933	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ 𝐴)
237	18	dmex 7758	. . . . . . . . . . . . 13 ⊢ dom 𝑐 ∈ V
238	237	elpw 4537	. . . . . . . . . . . 12 ⊢ (dom 𝑐 ∈ 𝒫 𝐴 ↔ dom 𝑐 ⊆ 𝐴)
239	236, 238	sylibr 233	. . . . . . . . . . 11 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ∈ 𝒫 𝐴)
240	224, 225, 239	3syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ 𝒫 𝐴)
241	223	elin2d 4133	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ Fin)
242		dmfi 9097	. . . . . . . . . . 11 ⊢ (𝑐 ∈ Fin → dom 𝑐 ∈ Fin)
243	241, 242	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ Fin)
244	240, 243	elind 4128	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ (𝒫 𝐴 ∩ Fin))
245		ovex 7308	. . . . . . . . . 10 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V
246	245	a1i 11	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V)
247		mpteq1 5167	. . . . . . . . . . 11 ⊢ (𝑎 = dom 𝑐 → (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶) = (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))
248	247	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = dom 𝑐 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
249	192, 248	elrnmpt1s 5866	. . . . . . . . 9 ⊢ ((dom 𝑐 ∈ (𝒫 𝐴 ∩ Fin) ∧ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
250	244, 246, 249	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
251		simpr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑡 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → 𝑡 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
252	251	breq2d 5086	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑡 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → (𝑠 < 𝑡 ↔ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
253		simpllr 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑠 ∈ ℝ^*)
254	11	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
255		nfcv 2907	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(ℝ^*_𝑠 ↾_s (0[,]+∞))
256		nfcv 2907	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧 Σ_g
257		nfmpt1 5182	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑐 ↦ 𝐹)
258	255, 256, 257	nfov 7305	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
259	110, 111, 258	nfbr 5121	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 𝑠 < ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
260	109, 259	nfan 1902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
261		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
262	260, 261	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
263		simp-4l 780	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝜑)
264	224, 225	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
265	264	sselda 3921	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
266	263, 265, 47	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝐹 ∈ (0[,]+∞))
267	266	ex 413	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (𝑧 ∈ 𝑐 → 𝐹 ∈ (0[,]+∞)))
268	262, 267	ralrimi 3141	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∀𝑧 ∈ 𝑐 𝐹 ∈ (0[,]+∞))
269	10, 254, 241, 268	gsummptcl 19568	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ (0[,]+∞))
270	9, 269	sselid 3919	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
271		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
272		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑐
273	25	nfpw 4554	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
274		nfcv 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗Fin
275	273, 274	nfin 4150	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
276	272, 275	nfel 2921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
277	271, 276	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
278		simpll 764	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝜑)
279	79, 236	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ⊆ 𝐴)
280	279	sselda 3921	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑗 ∈ 𝐴)
281	278, 280, 161	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
282	281	adantllr 716	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
283	282	adantllr 716	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
284	283	ex 413	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (𝑗 ∈ dom 𝑐 → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞)))
285	277, 284	ralrimi 3141	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∀𝑗 ∈ dom 𝑐Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
286	10, 254, 243, 285	gsummptcl 19568	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ (0[,]+∞))
287	9, 286	sselid 3919	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ ℝ^*)
288		simplr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑠 < ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
289	23, 276	nfan 1902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
290		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
291		xpss 5605	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑗} × 𝐵) ⊆ (V × V)
292	291	rgenw 3076	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V)
293		iunss 4975	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V) ↔ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V))
294	292, 293	mpbir 230	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V)
295	294	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V))
296	290, 295	sstrd 3931	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ (V × V))
297	79, 296	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ⊆ (V × V))
298		df-rel 5596	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑐 ↔ 𝑐 ⊆ (V × V))
299	297, 298	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Rel 𝑐)
300	29, 289, 10, 32, 299, 14, 12, 48	gsummpt2d 31309	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ (𝑐 “ {𝑗}) ↦ 𝐶)))))
301		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗dom 𝑐
302	237	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ V)
303	278	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝜑)
304	280	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝑗 ∈ 𝐴)
305	79	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
306		imass1 6009	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → (𝑐 “ {𝑗}) ⊆ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}))
307	305, 306	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ⊆ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}))
308	59, 61	iunsnima 30958	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}) = 𝐵)
309	278, 280, 308	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}) = 𝐵)
310	307, 309	sseqtrd 3961	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ⊆ 𝐵)
311	310	sselda 3921	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝑘 ∈ 𝐵)
312	303, 304, 311, 37	syl12anc 834	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝐶 ∈ (0[,]+∞))
313	312	ralrimiva 3103	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
314		imaexg 7762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ V → (𝑐 “ {𝑗}) ∈ V)
315	18, 314	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 “ {𝑗}) ∈ V
316		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑐 “ {𝑗})
317	316	esumcl 31998	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 “ {𝑗}) ∈ V ∧ ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
318	315, 317	mpan 687	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
319	313, 318	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
320		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐)
321	278, 280, 61	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝐵 ∈ 𝑊)
322	278	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝜑)
323	280	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝑗 ∈ 𝐴)
324		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
325	322, 323, 324, 37	syl12anc 834	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
326	320, 321, 325, 310	esummono 32022	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶 ≤ Σ^𝑘 ∈ 𝐵𝐶)
327	289, 301, 302, 319, 281, 326	esumlef 32030	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑗 ∈ dom 𝑐Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶 ≤ Σ^𝑗 ∈ dom 𝑐Σ^𝑘 ∈ 𝐵𝐶)
328	14, 242	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ Fin)
329	289, 301, 328, 319	esumgsum 32013	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑗 ∈ dom 𝑐Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶)))
330	14	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑐 ∈ Fin)
331		imafi2 31046	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ Fin → (𝑐 “ {𝑗}) ∈ Fin)
332	330, 331	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ∈ Fin)
333	320, 316, 332, 312	esumgsum 32013	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ (𝑐 “ {𝑗}) ↦ 𝐶)))
334	289, 333	mpteq2da 5172	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶) = (𝑗 ∈ dom 𝑐 ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ (𝑐 “ {𝑗}) ↦ 𝐶))))
335	334	oveq2d 7291	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ (𝑐 “ {𝑗}) ↦ 𝐶)))))
336	329, 335	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑗 ∈ dom 𝑐Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ (𝑐 “ {𝑗}) ↦ 𝐶)))))
337	289, 301, 328, 281	esumgsum 32013	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑗 ∈ dom 𝑐Σ^𝑘 ∈ 𝐵𝐶 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
338	327, 336, 337	3brtr3d 5105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ (𝑐 “ {𝑗}) ↦ 𝐶)))) ≤ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
339	300, 338	eqbrtrd 5096	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ≤ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^*𝑘 ∈ 𝐵𝐶)))
340	339	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ≤ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
341	340	adantlr 712	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ≤ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^*𝑘 ∈ 𝐵𝐶)))
342	253, 270, 287, 288, 341	xrltletrd 12895	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
343	250, 252, 342	rspcedvd 3563	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∃𝑡 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))𝑠 < 𝑡)
344	343	adantllr 716	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∃𝑡 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))𝑠 < 𝑡)
345	218, 221, 222, 344	syl21anc 835	. . . . 5 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → ∃𝑡 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))𝑠 < 𝑡)
346	53, 89	elrnmpti 5869	. . . . . . 7 ⊢ (𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ↔ ∃𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
347	346	biimpi 215	. . . . . 6 ⊢ (𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → ∃𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
348	347	ad2antlr 724	. . . . 5 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) → ∃𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
349	209, 345, 348	r19.29af 3262	. . . 4 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) → ∃𝑡 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^*𝑘 ∈ 𝐵𝐶)))𝑠 < 𝑡)
350	2, 134	suplub 9219	. . . . . 6 ⊢ (𝜑 → ((𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )) → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡))
351	350	imp 407	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)
352		breq2 5078	. . . . . 6 ⊢ (𝑡 = 𝑢 → (𝑠 < 𝑡 ↔ 𝑠 < 𝑢))
353	352	cbvrexvw 3384	. . . . 5 ⊢ (∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡 ↔ ∃𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑢)
354	351, 353	sylib 217	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) → ∃𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑢)
355	349, 354	r19.29a 3218	. . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) → ∃𝑡 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))𝑠 < 𝑡)
356	2, 135, 197, 355	eqsupd 9216	. 2 ⊢ (𝜑 → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))), ℝ^, < ) = sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^*, < ))
357		nfcv 2907	. . 3 ⊢ Ⅎ𝑗𝐴
358		eqidd 2739	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
359	23, 357, 59, 161, 358	esumval 32014	. 2 ⊢ (𝜑 → Σ^𝑗 ∈ 𝐴Σ^𝑘 ∈ 𝐵𝐶 = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))), ℝ^*, < ))
360	356, 359, 182	3eqtr4d 2788	1 ⊢ (𝜑 → Σ^𝑗 ∈ 𝐴Σ^𝑘 ∈ 𝐵𝐶 = Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Ⅎwnfc 2887 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 𝒫 cpw 4533 {csn 4561 ⟨cop 4567 ∪ ciun 4924 class class class wbr 5074 ↦ cmpt 5157 Or wor 5502 × cxp 5587 dom cdm 5589 ran crn 5590 “ cima 5592 Rel wrel 5594 (class class class)co 7275 Fincfn 8733 supcsup 9199 0cc0 10871 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 [,]cicc 13082 ↾_s cress 16941 Σ_g cgsu 17151 ℝ^*_𝑠cxrs 17211 CMndccmn 19386 Σ^*cesum 31995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-ordt 17212 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-ps 18284 df-tsr 18285 df-plusf 18325 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-subrg 20022 df-abv 20077 df-lmod 20125 df-scaf 20126 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-tmd 23223 df-tgp 23224 df-tsms 23278 df-trg 23311 df-xms 23473 df-ms 23474 df-tms 23475 df-nm 23738 df-ngp 23739 df-nrg 23741 df-nlm 23742 df-ii 24040 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-esum 31996

This theorem is referenced by: esumiun 32062

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