esum2d - Mathbox for Thierry Arnoux

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Theorem esum2d 29482

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 11850	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑐𝜑
4		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑐𝑠
5		nfmpt1 4675	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
6	5	nfrn 5289	. . . . . . . . . 10 ⊢ Ⅎ𝑐ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
7	4, 6	nfel 2763	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
8	3, 7	nfan 1816	. . . . . . . 8 ⊢ Ⅎ𝑐(𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
9		iccssxr 12127	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ^*
10		xrge0base 29016	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11		xrge0cmn 19607	. . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
14	13	elin2d 3765	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ Fin)
15		simpll 786	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝜑)
16	13	elin1d 3764	. . . . . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑐 ∈ V
19	18	elpw 4114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
20	17, 19	sylib 207	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
21		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ 𝑐)
22	20, 21	sseldd 3569	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
23		nfv 1830	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗𝜑
24		nfcv 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗𝑧
25		nfiu1 4486	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
26	24, 25	nfel 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
27	23, 26	nfan 1816	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
28		nfv 1830	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵))
29		esum2d.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘𝐹
30		nfcv 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(0[,]+∞)
31	29, 30	nfel 2763	. . . . . . . . . . . . . . . . . . 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 804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝜑)
35		simp-4r 803	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝑗 ∈ 𝐴)
36		simplr 788	. . . . . . . . . . . . . . . . . . . . 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 1316	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐶 ∈ (0[,]+∞))
39	33, 38	eqeltrd 2688	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 ∈ (0[,]+∞))
40		elsnxp 5594	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝐴 → (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩))
41	40	biimpa 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
42	41	adantll 746	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
43	28, 31, 39, 42	r19.29af2 3057	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
44		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
45		eliun 4460	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
46	44, 45	sylib 207	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
47	27, 43, 46	r19.29af 3058	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
48	15, 22, 47	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝐹 ∈ (0[,]+∞))
49	48	ralrimiva 2949	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∀𝑧 ∈ 𝑐 𝐹 ∈ (0[,]+∞))
50	10, 12, 14, 49	gsummptcl 18189	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ (0[,]+∞))
51	9, 50	sseldi 3566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
52	51	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
53		eqid 2610	. . . . . . . . . . . . 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 6299	. . . . . . . . . . . 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 762	. . . . . . . . . 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 795	. . . . . . . . . 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 3569	. . . . . . . . 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 4835	. . . . . . . . . . . . . . 15 ⊢ {𝑗} ∈ V
61		esum2d.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
62		xpexg 6858	. . . . . . . . . . . . . . 15 ⊢ (({𝑗} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝑗} × 𝐵) ∈ V)
63	60, 61, 62	sylancr 694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
64	63	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
65		iunexg 7035	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
66	59, 64, 65	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
67	47	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
68		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
69	68	esumcl 29419	. . . . . . . . . . . 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 691	. . . . . . . . . . 11 ⊢ (𝜑 → Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
71	9, 70	sseldi 3566	. . . . . . . . . 10 ⊢ (𝜑 → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
72	71	ad3antrrr 762	. . . . . . . . 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 476	. . . . . . . . . 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 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
75		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝑐
76	74, 75, 14, 48	esumgsum 29434	. . . . . . . . . . . . 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 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
79	16, 19	sylib 207	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
80	74, 77, 78, 79	esummono 29443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
81	76, 80	eqbrtrrd 4607	. . . . . . . . . . . 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 747	. . . . . . . . . . 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 4605	. . . . . . . . 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 9982	. . . . . . . . . 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 500	. . . . . . . . 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 1317	. . . . . . . 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 476	. . . . . . . . 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 6577	. . . . . . . . . 10 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ V
90	53, 89	elrnmpti 5297	. . . . . . . . 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 207	. . . . . . . 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 3058	. . . . . . 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 2949	. . . . . 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 1830	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
95		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑐 𝑠 < 𝑡
96	6, 95	nfrex 2990	. . . . . . . . 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 747	. . . . . . . . . . . . 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 747	. . . . . . . . . . . 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 788	. . . . . . . . . . . 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 5295	. . . . . . . . . . . 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 691	. . . . . . . . . . 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 2688	. . . . . . . . . 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 476	. . . . . . . . . . 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 4595	. . . . . . . . . 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 476	. . . . . . . . . 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 3289	. . . . . . . . 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 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑠 ∈ ℝ^*)
110		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝑠
111		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧 <
112	68	nfesum1 29429	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
113	110, 111, 112	nfbr 4629	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
114	109, 113	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
115	66	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
116	47	3ad2antr3 1221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → 𝐹 ∈ (0[,]+∞))
117	116	3anassrs 1282	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
118		simplr 788	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑠 ∈ ℝ^*)
119		simpr 476	. . . . . . . . . 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 29449	. . . . . . . . 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 3057	. . . . . . . 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 449	. . . . . . 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 2949	. . . . . 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 553	. . . . 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 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
126	125	breq1d 4593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (𝑟 < 𝑠 ↔ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
127	126	notbid 307	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (¬ 𝑟 < 𝑠 ↔ ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
128	127	ralbidv 2969	. . . . . . 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 4595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (𝑠 < 𝑟 ↔ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹))
130	129	imbi1d 330	. . . . . . . 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 2969	. . . . . . 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 743	. . . . . 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 3286	. . . . 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 8247	. . 3 ⊢ (𝜑 → sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) ∈ ℝ^*)
136		nfv 1830	. . . . 5 ⊢ Ⅎ𝑎𝜑
137		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑎𝑠
138		nfmpt1 4675	. . . . . . 7 ⊢ Ⅎ𝑎(𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
139	138	nfrn 5289	. . . . . 6 ⊢ Ⅎ𝑎ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
140	137, 139	nfel 2763	. . . . 5 ⊢ Ⅎ𝑎 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
141	136, 140	nfan 1816	. . . 4 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
142		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
143		simpll 786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝜑)
144	142	elin1d 3764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ 𝒫 𝐴)
145		elpwi 4117	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
146	144, 145	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ⊆ 𝐴)
147	146	sselda 3568	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝑗 ∈ 𝐴)
148	143, 147, 61	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝐵 ∈ 𝑊)
149	143	adantrr 749	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝜑)
150	147	adantrr 749	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑗 ∈ 𝐴)
151		simprr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑘 ∈ 𝐵)
152	149, 150, 151, 37	syl12anc 1316	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
153	142	elin2d 3765	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ Fin)
154	29, 32, 142, 148, 152, 153	esum2dlem 29481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑗 ∈ 𝑎Σ^𝑘 ∈ 𝐵𝐶 = Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹)
155		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
156		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑎
157	37	anassrs 678	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
158	157	ralrimiva 2949	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
159		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐵
160	159	esumcl 29419	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
161	61, 158, 160	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
162	143, 147, 161	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
163	155, 156, 153, 162	esumgsum 29434	. . . . . . . . . . 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 2646	. . . . . . . . . 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 1830	. . . . . . . . . . 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 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
168		iunss1 4468	. . . . . . . . . . . 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 29443	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
171	164, 170	eqbrtrrd 4607	. . . . . . . . 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 2949	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑗 ∈ 𝑎 Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
174	10, 172, 153, 173	gsummptcl 18189	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ (0[,]+∞))
175	9, 174	sseldi 3566	. . . . . . . . . 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 9982	. . . . . . . . . 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 691	. . . . . . . . 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 221	. . . . . . . 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 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝜑
181		eqidd 2611	. . . . . . . . . . 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 29435	. . . . . . . . . 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 4593	. . . . . . . 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 313	. . . . . . 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 747	. . . . . 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 476	. . . . . . 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 4595	. . . . . 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 307	. . . . 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 246	. . . 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 2610	. . . . . . 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 6577	. . . . . . 7 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V
194	192, 193	elrnmpti 5297	. . . . . 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 205	. . . . 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 3058	. . 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 2765	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ℝ^*
199		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑐 <
200		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑐ℝ^*
201	6, 200, 199	nfsup 8240	. . . . . . . . . 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 4629	. . . . . . . . 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 1816	. . . . . . . 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 1816	. . . . . . 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 2751	. . . . . . . 8 ⊢ Ⅎ𝑐𝑢
206	205, 6	nfel 2763	. . . . . . 7 ⊢ Ⅎ𝑐 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
207	204, 206	nfan 1816	. . . . . 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 1830	. . . . . 6 ⊢ Ⅎ𝑐 𝑠 < 𝑢
209	207, 208	nfan 1816	. . . . 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 804	. . . . . . . 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 1111	. . . . . . . . . 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 1282	. . . . . . . . 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 1282	. . . . . . . 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 553	. . . . . . 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 1112	. . . . . . . . 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 1282	. . . . . . . 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 1282	. . . . . . 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 553	. . . . . 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 795	. . . . . . 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 476	. . . . . . 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 4609	. . . . . 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 788	. . . . . 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 476	. . . . . . . . . . . 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 3764	. . . . . . . . . . 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 4117	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
226		dmss 5245	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
227		dmiun 5255	. . . . . . . . . . . . . 14 ⊢ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵)
228	226, 227	syl6sseq 3614	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵))
229		dmxpss 5484	. . . . . . . . . . . . . . . . 17 ⊢ dom ({𝑗} × 𝐵) ⊆ {𝑗}
230	229	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ {𝑗})
231		snssi 4280	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → {𝑗} ⊆ 𝐴)
232	230, 231	sstrd 3578	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ 𝐴)
233	232	rgen 2906	. . . . . . . . . . . . . 14 ⊢ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
234		iunss 4497	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴 ↔ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴)
235	233, 234	mpbir 220	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
236	228, 235	syl6ss 3580	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ 𝐴)
237	18	dmex 6991	. . . . . . . . . . . . 13 ⊢ dom 𝑐 ∈ V
238	237	elpw 4114	. . . . . . . . . . . 12 ⊢ (dom 𝑐 ∈ 𝒫 𝐴 ↔ dom 𝑐 ⊆ 𝐴)
239	236, 238	sylibr 223	. . . . . . . . . . 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 3765	. . . . . . . . . . 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 8129	. . . . . . . . . . 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 3760	. . . . . . . . 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 6577	. . . . . . . . . 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 4665	. . . . . . . . . . 11 ⊢ (𝑎 = dom 𝑐 → (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶) = (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))
248	247	oveq2d 6565	. . . . . . . . . 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 5294	. . . . . . . . 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 691	. . . . . . . 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 476	. . . . . . . . 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 4595	. . . . . . . 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 795	. . . . . . . . 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 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(ℝ^*_𝑠 ↾_s (0[,]+∞))
256		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧 Σ_g
257		nfmpt1 4675	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑐 ↦ 𝐹)
258	255, 256, 257	nfov 6575	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
259	110, 111, 258	nfbr 4629	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 𝑠 < ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
260	109, 259	nfan 1816	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
261		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
262	260, 261	nfan 1816	. . . . . . . . . . . 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 802	. . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . . 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 691	. . . . . . . . . . . . 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 449	. . . . . . . . . . . 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 2940	. . . . . . . . . . 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 18189	. . . . . . . . . 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	sseldi 3566	. . . . . . . . 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 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
272		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑐
273	25	nfpw 4120	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
274		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗Fin
275	273, 274	nfin 3782	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
276	272, 275	nfel 2763	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
277	271, 276	nfan 1816	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
278		simpll 786	. . . . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑗 ∈ 𝐴)
281	278, 280, 161	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
282	281	adantllr 751	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
283	282	adantllr 751	. . . . . . . . . . . . 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 449	. . . . . . . . . . . 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 2940	. . . . . . . . . . 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 18189	. . . . . . . . . 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	sseldi 3566	. . . . . . . . 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 788	. . . . . . . . 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 1816	. . . . . . . . . . . . 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 5149	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑗} × 𝐵) ⊆ (V × V)
292	291	rgenw 2908	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V)
293		iunss 4497	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V) ↔ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V))
294	292, 293	mpbir 220	. . . . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . . 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 5045	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑐 ↔ 𝑐 ⊆ (V × V))
299	297, 298	sylibr 223	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Rel 𝑐)
300	29, 289, 10, 32, 299, 14, 12, 48	gsummpt2d 29112	. . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 5419	. . . . . . . . . . . . . . . . . . . 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 28808	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}) = 𝐵)
309	278, 280, 308	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}) = 𝐵)
310	307, 309	sseqtrd 3604	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ⊆ 𝐵)
311	310	sselda 3568	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝑘 ∈ 𝐵)
312	303, 304, 311, 37	syl12anc 1316	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝐶 ∈ (0[,]+∞))
313	312	ralrimiva 2949	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
314		imaexg 6995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ V → (𝑐 “ {𝑗}) ∈ V)
315	18, 314	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 “ {𝑗}) ∈ V
316		nfcv 2751	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑐 “ {𝑗})
317	316	esumcl 29419	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 “ {𝑗}) ∈ V ∧ ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
318	315, 317	mpan 702	. . . . . . . . . . . . . . 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 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐)
321	278, 280, 61	syl2anc 691	. . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
325	322, 323, 324, 37	syl12anc 1316	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
326	320, 321, 325, 310	esummono 29443	. . . . . . . . . . . . . 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 29451	. . . . . . . . . . . . 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 29434	. . . . . . . . . . . . . 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 28872	. . . . . . . . . . . . . . . . . 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 29434	. . . . . . . . . . . . . . . 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 4671	. . . . . . . . . . . . . . 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 6565	. . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . 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 29434	. . . . . . . . . . . . 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 4614	. . . . . . . . . . . 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 4605	. . . . . . . . . . 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 747	. . . . . . . . . 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 747	. . . . . . . . 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 11868	. . . . . . . 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 3289	. . . . . . 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 751	. . . . . 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 1317	. . . . 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 5297	. . . . . . 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 205	. . . . . 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 759	. . . . 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 3058	. . . 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 8249	. . . . . 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 444	. . . . 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 4587	. . . . . 6 ⊢ (𝑡 = 𝑢 → (𝑠 < 𝑡 ↔ 𝑠 < 𝑢))
353	352	cbvrexv 3148	. . . . 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 207	. . . 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 3060	. . 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 8246	. 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 2751	. . 3 ⊢ Ⅎ𝑗𝐴
358		eqidd 2611	. . 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 29435	. 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 2654	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ⟨cop 4131 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 Or wor 4958 × cxp 5036 dom cdm 5038 ran crn 5039 “ cima 5041 Rel wrel 5043 (class class class)co 6549 Fincfn 7841 supcsup 8229 0cc0 9815 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 [,]cicc 12049 ↾_s cress 15696 Σ_g cgsu 15924 ℝ^*_𝑠cxrs 15983 CMndccmn 18016 Σ^*cesum 29416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-esum 29417

This theorem is referenced by: esumiun 29483

W3C validator