esum2d - Mathbox for Thierry Arnoux

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Theorem esum2d 34070

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 13155	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑐𝜑
4		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑐𝑠
5		nfmpt1 5220	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
6	5	nfrn 5932	. . . . . . . . . 10 ⊢ Ⅎ𝑐ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
7	4, 6	nfel 2913	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
8	3, 7	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑐(𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
9		iccssxr 13445	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ^*
10		xrge0base 32952	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11		xrge0cmn 21374	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
14	13	elin2d 4180	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ Fin)
15		simpll 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝜑)
16	13	elin1d 4179	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
17	16	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
18		vex 3463	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑐 ∈ V
19	18	elpw 4579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
20	17, 19	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
21		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ 𝑐)
22	20, 21	sseldd 3959	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
23		nfv 1914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗𝜑
24		nfcv 2898	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗𝑧
25		nfiu1 5003	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
26	24, 25	nfel 2913	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
27	23, 26	nfan 1899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
28		nfv 1914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵))
29		esum2d.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘𝐹
30		nfcv 2898	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(0[,]+∞)
31	29, 30	nfel 2913	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘 𝐹 ∈ (0[,]+∞)
32		esum2d.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)
33	32	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 = 𝐶)
34		simp-5l 784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝜑)
35		simp-4r 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝑗 ∈ 𝐴)
36		simplr 768	. . . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐶 ∈ (0[,]+∞))
39	33, 38	eqeltrd 2834	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 ∈ (0[,]+∞))
40		elsnxp 6280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝐴 → (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩))
41	40	biimpa 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
42	41	adantll 714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
43	28, 31, 39, 42	r19.29af2 3250	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
44		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
45		eliun 4971	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
46	44, 45	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
47	27, 43, 46	r19.29af 3251	. . . . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∀𝑧 ∈ 𝑐 𝐹 ∈ (0[,]+∞))
50	10, 12, 14, 49	gsummptcl 19946	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ (0[,]+∞))
51	9, 50	sselid 3956	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
52	51	ralrimiva 3132	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
53		eqid 2735	. . . . . . . . . . . . 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 7112	. . . . . . . . . . . 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 730	. . . . . . . . . 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 775	. . . . . . . . . 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 3959	. . . . . . . . 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		vsnex 5404	. . . . . . . . . . . . . . 15 ⊢ {𝑗} ∈ V
61		esum2d.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
62		xpexg 7742	. . . . . . . . . . . . . . 15 ⊢ (({𝑗} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝑗} × 𝐵) ∈ V)
63	60, 61, 62	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
64	63	ralrimiva 3132	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
65		iunexg 7960	. . . . . . . . . . . . 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 3132	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
68		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
69	68	esumcl 34007	. . . . . . . . . . . 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 3956	. . . . . . . . . 10 ⊢ (𝜑 → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
72	71	ad3antrrr 730	. . . . . . . . 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 484	. . . . . . . . . 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 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
75		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝑐
76	74, 75, 14, 48	esumgsum 34022	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
77	66	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
78	47	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
79	16, 19	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
80	74, 77, 78, 79	esummono 34031	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
81	76, 80	eqbrtrrd 5143	. . . . . . . . . . . 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 715	. . . . . . . . . . 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 480	. . . . . . . . . 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 5141	. . . . . . . . 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 11298	. . . . . . . . . 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 476	. . . . . . . . 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 837	. . . . . . . 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 484	. . . . . . . . 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 7436	. . . . . . . . . 10 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ V
90	53, 89	elrnmpti 5942	. . . . . . . . 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 218	. . . . . . . 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 3251	. . . . . . 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 3132	. . . . . 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 1914	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
95		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑐 𝑠 < 𝑡
96	6, 95	nfrexw 3293	. . . . . . . . 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 715	. . . . . . . . . . . . 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 715	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 768	. . . . . . . . . . . 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 5940	. . . . . . . . . . . 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 2834	. . . . . . . . . 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 484	. . . . . . . . . . 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 5131	. . . . . . . . . 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 484	. . . . . . . . . 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 3603	. . . . . . . . 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 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑠 ∈ ℝ^*)
110		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝑠
111		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧 <
112	68	nfesum1 34017	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
113	110, 111, 112	nfbr 5166	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
114	109, 113	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
115	66	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
116	47	3ad2antr3 1191	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → 𝐹 ∈ (0[,]+∞))
117	116	3anassrs 1361	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
118		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑠 ∈ ℝ^*)
119		simpr 484	. . . . . . . . . 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 34037	. . . . . . . . 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 3250	. . . . . . . 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 412	. . . . . . 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 3132	. . . . . 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 511	. . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
126	125	breq1d 5129	. . . . . . . . 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 3163	. . . . . . 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 5131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (𝑠 < 𝑟 ↔ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹))
130	129	imbi1d 341	. . . . . . . 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 3163	. . . . . . 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 632	. . . . . 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 3594	. . . . 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 9468	. . 3 ⊢ (𝜑 → sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) ∈ ℝ^*)
136		nfv 1914	. . . . 5 ⊢ Ⅎ𝑎𝜑
137		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑎𝑠
138		nfmpt1 5220	. . . . . . 7 ⊢ Ⅎ𝑎(𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
139	138	nfrn 5932	. . . . . 6 ⊢ Ⅎ𝑎ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
140	137, 139	nfel 2913	. . . . 5 ⊢ Ⅎ𝑎 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
141	136, 140	nfan 1899	. . . 4 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
142		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
143		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝜑)
144	142	elin1d 4179	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ 𝒫 𝐴)
145		elpwi 4582	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
146	144, 145	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ⊆ 𝐴)
147	146	sselda 3958	. . . . . . . . . . . . 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 717	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝜑)
150	147	adantrr 717	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑗 ∈ 𝐴)
151		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑘 ∈ 𝐵)
152	149, 150, 151, 37	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
153	142	elin2d 4180	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ Fin)
154	29, 32, 142, 148, 152, 153	esum2dlem 34069	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑗 ∈ 𝑎Σ^𝑘 ∈ 𝐵𝐶 = Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹)
155		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
156		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑎
157	37	anassrs 467	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
158	157	ralrimiva 3132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
159		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐵
160	159	esumcl 34007	. . . . . . . . . . . . . 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 34022	. . . . . . . . . . 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 2772	. . . . . . . . . 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 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
166	66	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
167	47	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
168		iunss1 4982	. . . . . . . . . . . 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 34031	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
171	164, 170	eqbrtrrd 5143	. . . . . . . . 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 3132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑗 ∈ 𝑎 Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
174	10, 172, 153, 173	gsummptcl 19946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ (0[,]+∞))
175	9, 174	sselid 3956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ ℝ^*)
176	71	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
177		xrlenlt 11298	. . . . . . . . . 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 232	. . . . . . . 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 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝜑
181		eqidd 2736	. . . . . . . . . . 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 34023	. . . . . . . . . 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 480	. . . . . . . . 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 5129	. . . . . . . 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 715	. . . . . 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 480	. . . . 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 484	. . . . . . 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 5131	. . . . . 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 257	. . . 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 2735	. . . . . . 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 7436	. . . . . . 7 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V
194	192, 193	elrnmpti 5942	. . . . . 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 216	. . . . 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 481	. . . 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 3251	. . 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 2915	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ℝ^*
199		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑐 <
200		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑐ℝ^*
201	6, 200, 199	nfsup 9461	. . . . . . . . . 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 5166	. . . . . . . . 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 1899	. . . . . . . 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 1899	. . . . . . 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 2898	. . . . . . . 8 ⊢ Ⅎ𝑐𝑢
206	205, 6	nfel 2913	. . . . . . 7 ⊢ Ⅎ𝑐 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
207	204, 206	nfan 1899	. . . . . 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 1914	. . . . . 6 ⊢ Ⅎ𝑐 𝑠 < 𝑢
209	207, 208	nfan 1899	. . . . 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 784	. . . . . . . 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 1231	. . . . . . . . . 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 1361	. . . . . . . . 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 1361	. . . . . . . 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 511	. . . . . . 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 1232	. . . . . . . . 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 1361	. . . . . . . 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 1361	. . . . . . 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 511	. . . . . 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 775	. . . . . . 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 484	. . . . . . 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 5145	. . . . . 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 768	. . . . . 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 484	. . . . . . . . . . . 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 4179	. . . . . . . . . . 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 4582	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
226		dmss 5882	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
227		dmiun 5893	. . . . . . . . . . . . . 14 ⊢ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵)
228	226, 227	sseqtrdi 3999	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵))
229		dmxpss 6160	. . . . . . . . . . . . . . . . 17 ⊢ dom ({𝑗} × 𝐵) ⊆ {𝑗}
230	229	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ {𝑗})
231		snssi 4784	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → {𝑗} ⊆ 𝐴)
232	230, 231	sstrd 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ 𝐴)
233	232	rgen 3053	. . . . . . . . . . . . . 14 ⊢ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
234		iunss 5021	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴 ↔ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴)
235	233, 234	mpbir 231	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
236	228, 235	sstrdi 3971	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ 𝐴)
237	18	dmex 7903	. . . . . . . . . . . . 13 ⊢ dom 𝑐 ∈ V
238	237	elpw 4579	. . . . . . . . . . . 12 ⊢ (dom 𝑐 ∈ 𝒫 𝐴 ↔ dom 𝑐 ⊆ 𝐴)
239	236, 238	sylibr 234	. . . . . . . . . . 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 4180	. . . . . . . . . . 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 9345	. . . . . . . . . . 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 4175	. . . . . . . . 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 7436	. . . . . . . . . 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 5209	. . . . . . . . . . 11 ⊢ (𝑎 = dom 𝑐 → (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶) = (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))
248	247	oveq2d 7419	. . . . . . . . . 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 5939	. . . . . . . . 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 484	. . . . . . . . 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 5131	. . . . . . . 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 775	. . . . . . . . 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 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(ℝ^*_𝑠 ↾_s (0[,]+∞))
256		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧 Σ_g
257		nfmpt1 5220	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑐 ↦ 𝐹)
258	255, 256, 257	nfov 7433	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
259	110, 111, 258	nfbr 5166	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 𝑠 < ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
260	109, 259	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
261		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
262	260, 261	nfan 1899	. . . . . . . . . . . 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 782	. . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3240	. . . . . . . . . . 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 19946	. . . . . . . . . 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 3956	. . . . . . . . 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 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
272		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑐
273	25	nfpw 4594	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
274		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗Fin
275	273, 274	nfin 4199	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
276	272, 275	nfel 2913	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
277	271, 276	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
278		simpll 766	. . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
283	282	adantllr 719	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3240	. . . . . . . . . . 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 19946	. . . . . . . . . 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 3956	. . . . . . . . 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 768	. . . . . . . . 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 1899	. . . . . . . . . . . . 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 5670	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑗} × 𝐵) ⊆ (V × V)
292	291	rgenw 3055	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V)
293		iunss 5021	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V) ↔ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V))
294	292, 293	mpbir 231	. . . . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . . 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 5661	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑐 ↔ 𝑐 ⊆ (V × V))
299	297, 298	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Rel 𝑐)
300	29, 289, 10, 32, 299, 14, 12, 48	gsummpt2d 32989	. . . . . . . . . . . 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 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗dom 𝑐
302	237	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ V)
303	278	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝜑)
304	280	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝑗 ∈ 𝐴)
305	79	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
306		imass1 6088	. . . . . . . . . . . . . . . . . . . 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 32544	. . . . . . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ⊆ 𝐵)
311	310	sselda 3958	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝑘 ∈ 𝐵)
312	303, 304, 311, 37	syl12anc 836	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝐶 ∈ (0[,]+∞))
313	312	ralrimiva 3132	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
314		imaexg 7907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ V → (𝑐 “ {𝑗}) ∈ V)
315	18, 314	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 “ {𝑗}) ∈ V
316		nfcv 2898	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑐 “ {𝑗})
317	316	esumcl 34007	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 “ {𝑗}) ∈ V ∧ ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
318	315, 317	mpan 690	. . . . . . . . . . . . . . 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 1914	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝜑)
323	280	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝑗 ∈ 𝐴)
324		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
325	322, 323, 324, 37	syl12anc 836	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
326	320, 321, 325, 310	esummono 34031	. . . . . . . . . . . . . 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 34039	. . . . . . . . . . . . 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 34022	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑐 ∈ Fin)
331		imafi2 32635	. . . . . . . . . . . . . . . . . 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 34022	. . . . . . . . . . . . . . . 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 5213	. . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . 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 34022	. . . . . . . . . . . . 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 5150	. . . . . . . . . . . 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 5141	. . . . . . . . . . 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 715	. . . . . . . . . 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 715	. . . . . . . . 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 13175	. . . . . . . 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 3603	. . . . . . 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 719	. . . . . 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 837	. . . . 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 5942	. . . . . . 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 216	. . . . . 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 727	. . . . 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 3251	. . . 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 9470	. . . . . 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 406	. . . . 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 5123	. . . . . 6 ⊢ (𝑡 = 𝑢 → (𝑠 < 𝑡 ↔ 𝑠 < 𝑢))
353	352	cbvrexvw 3221	. . . . 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 218	. . . 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 3148	. . 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 9467	. 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 2898	. . 3 ⊢ Ⅎ𝑗𝐴
358		eqidd 2736	. . 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 34023	. 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 2780	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2883 ∀wral 3051 ∃wrex 3060 Vcvv 3459 ∩ cin 3925 ⊆ wss 3926 𝒫 cpw 4575 {csn 4601 ⟨cop 4607 ∪ ciun 4967 class class class wbr 5119 ↦ cmpt 5201 Or wor 5560 × cxp 5652 dom cdm 5654 ran crn 5655 “ cima 5657 Rel wrel 5659 (class class class)co 7403 Fincfn 8957 supcsup 9450 0cc0 11127 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 [,]cicc 13363 ↾_s cress 17249 Σ_g cgsu 17452 ℝ^*_𝑠cxrs 17512 CMndccmn 19759 Σ^*cesum 34004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-ef 16081 df-sin 16083 df-cos 16084 df-pi 16086 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-ordt 17513 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-ps 18574 df-tsr 18575 df-plusf 18615 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-mhm 18759 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-mulg 19049 df-subg 19104 df-cntz 19298 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-cring 20194 df-subrng 20504 df-subrg 20528 df-abv 20767 df-lmod 20817 df-scaf 20818 df-sra 21129 df-rgmod 21130 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-tmd 24008 df-tgp 24009 df-tsms 24063 df-trg 24096 df-xms 24257 df-ms 24258 df-tms 24259 df-nm 24519 df-ngp 24520 df-nrg 24522 df-nlm 24523 df-ii 24819 df-cncf 24820 df-limc 25817 df-dv 25818 df-log 26515 df-esum 34005

This theorem is referenced by: esumiun 34071

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