esum2d - Mathbox for Thierry Arnoux

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Theorem esum2d 34277

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 13083	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑐𝜑
4		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑐𝑠
5		nfmpt1 5171	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
6	5	nfrn 5894	. . . . . . . . . 10 ⊢ Ⅎ𝑐ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
7	4, 6	nfel 2915	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
8	3, 7	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑐(𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
9		iccssxr 13374	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ^*
10		xrge0base 17562	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11		xrge0cmn 21419	. . . . . . . . . . . . . . . 16 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
12	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
13		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
14	13	elin2d 4134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ Fin)
15		simpll 772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝜑)
16	13	elin1d 4133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
17	16	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
18		vex 3435	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑐 ∈ V
19	18	elpw 4533	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
20	17, 19	sylib 219	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
21		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ 𝑐)
22	20, 21	sseldd 3916	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
23		nfv 1921	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗𝜑
24		nfcv 2901	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗𝑧
25		nfiu1 4957	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
26	24, 25	nfel 2915	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
27	23, 26	nfan 1906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
28		nfv 1921	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵))
29		esum2d.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘𝐹
30		nfcv 2901	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(0[,]+∞)
31	29, 30	nfel 2915	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘 𝐹 ∈ (0[,]+∞)
32		esum2d.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)
33	32	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 = 𝐶)
34		simp-5l 790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝜑)
35		simp-4r 789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝑗 ∈ 𝐴)
36		simplr 774	. . . . . . . . . . . . . . . . . . . . 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 842	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐶 ∈ (0[,]+∞))
39	33, 38	eqeltrd 2839	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 ∈ (0[,]+∞))
40		elsnxp 6242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝐴 → (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩))
41	40	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
42	41	adantll 720	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
43	28, 31, 39, 42	r19.29af2 3247	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
44		eliun 4925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
45	44	bilani 505	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
46	27, 43, 45	r19.29af 3248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
47	15, 22, 46	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝐹 ∈ (0[,]+∞))
48	47	ralrimiva 3131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∀𝑧 ∈ 𝑐 𝐹 ∈ (0[,]+∞))
49	10, 12, 14, 48	gsummptcl 19933	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ (0[,]+∞))
50	9, 49	sselid 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
51	50	ralrimiva 3131	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ ℝ^)
52		eqid 2739	. . . . . . . . . . . . 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)))
53	52	rnmptss 7064	. . . . . . . . . . . 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^)
54	51, 53	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ⊆ ℝ^)
55	54	ad3antrrr 736	. . . . . . . . . 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^)
56		simpllr 781	. . . . . . . . . 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))))
57	55, 56	sseldd 3916	. . . . . . . . 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^*)
58		esum2d.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
59		vsnex 5364	. . . . . . . . . . . . . . 15 ⊢ {𝑗} ∈ V
60		esum2d.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
61		xpexg 7693	. . . . . . . . . . . . . . 15 ⊢ (({𝑗} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝑗} × 𝐵) ∈ V)
62	59, 60, 61	sylancr 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
63	62	ralrimiva 3131	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
64		iunexg 7905	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
65	58, 63, 64	syl2anc 590	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
66	46	ralrimiva 3131	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
67		nfcv 2901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
68	67	esumcl 34214	. . . . . . . . . . . 12 ⊢ ((∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V ∧ ∀𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞)) → Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
69	65, 66, 68	syl2anc 590	. . . . . . . . . . 11 ⊢ (𝜑 → Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ (0[,]+∞))
70	9, 69	sselid 3913	. . . . . . . . . 10 ⊢ (𝜑 → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
71	70	ad3antrrr 736	. . . . . . . . 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^)
72		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑠 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
73		nfv 1921	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
74		nfcv 2901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝑐
75	73, 74, 14, 47	esumgsum 34229	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
76	65	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
77	46	adantlr 721	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
78	16, 19	sylib 219	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
79	73, 76, 77, 78	esummono 34238	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
80	75, 79	eqbrtrrd 5096	. . . . . . . . . . . 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)
81	80	adantlr 721	. . . . . . . . . . 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)
82	81	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
83	72, 82	eqbrtrd 5094	. . . . . . . . 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)
84		xrlenlt 11201	. . . . . . . . . 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))
85	84	biimpa 477	. . . . . . . . 9 ⊢ (((𝑠 ∈ ℝ^* ∧ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^) ∧ 𝑠 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠)
86	57, 71, 83, 85	syl21anc 843	. . . . . . . 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)
87		ovex 7389	. . . . . . . . . 10 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)) ∈ V
88	52, 87	elrnmpti 5904	. . . . . . . . 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)))
89	88	bilani 505	. . . . . . . 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)))
90	8, 86, 89	r19.29af 3248	. . . . . . 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)
91	90	ralrimiva 3131	. . . . . 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)
92		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
93		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑐 𝑠 < 𝑡
94	6, 93	nfrexw 3287	. . . . . . . . 9 ⊢ Ⅎ𝑐∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡
95	75	adantlr 721	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
96	95	adantlr 721	. . . . . . . . . . . 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)))
97	96	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → Σ^𝑧 ∈ 𝑐𝐹 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
98		simplr 774	. . . . . . . . . . . 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))
99	87	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)
100	52	elrnmpt1 5902	. . . . . . . . . . . 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))))
101	98, 99, 100	syl2anc 590	. . . . . . . . . . 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))))
102	97, 101	eqeltrd 2839	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → Σ^𝑧 ∈ 𝑐𝐹 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))))
103		simpr 485	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) ∧ 𝑡 = Σ^𝑧 ∈ 𝑐𝐹) → 𝑡 = Σ^*𝑧 ∈ 𝑐𝐹)
104	103	breq2d 5084	. . . . . . . . . 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))
105		simpr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑠 < Σ^𝑧 ∈ 𝑐𝐹) → 𝑠 < Σ^𝑧 ∈ 𝑐𝐹)
106	102, 104, 105	rspcedvd 3562	. . . . . . . . 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)
107		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑠 ∈ ℝ^*)
108		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝑠
109		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧 <
110	67	nfesum1 34224	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
111	108, 109, 110	nfbr 5119	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹
112	107, 111	nfan 1906	. . . . . . . . . 10 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
113	65	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
114	46	3ad2antr3 1197	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → 𝐹 ∈ (0[,]+∞))
115	114	3anassrs 1367	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
116		simplr 774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑠 ∈ ℝ^*)
117		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑠 < Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
118	112, 113, 115, 116, 117	esumlub 34244	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ∃𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)𝑠 < Σ^*𝑧 ∈ 𝑐𝐹)
119	92, 94, 106, 118	r19.29af2 3247	. . . . . . . 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)
120	119	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ^) → (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡))
121	120	ralrimiva 3131	. . . . . 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))
122	91, 121	jca 516	. . . . 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)))
123		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
124	123	breq1d 5082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (𝑟 < 𝑠 ↔ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
125	124	notbid 319	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (¬ 𝑟 < 𝑠 ↔ ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < 𝑠))
126	125	ralbidv 3162	. . . . . . 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))
127	123	breq2d 5084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → (𝑠 < 𝑟 ↔ 𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹))
128	127	imbi1d 342	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) → ((𝑠 < 𝑟 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡) ↔ (𝑠 < Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)))
129	128	ralbidv 3162	. . . . . . 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)))
130	126, 129	anbi12d 638	. . . . . 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))))
131	70, 130	rspcedv 3553	. . . . 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))))
132	122, 131	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)))
133	2, 132	supcl 9361	. . 3 ⊢ (𝜑 → sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) ∈ ℝ^*)
134		nfv 1921	. . . . 5 ⊢ Ⅎ𝑎𝜑
135		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑎𝑠
136		nfmpt1 5171	. . . . . . 7 ⊢ Ⅎ𝑎(𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
137	136	nfrn 5894	. . . . . 6 ⊢ Ⅎ𝑎ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
138	135, 137	nfel 2915	. . . . 5 ⊢ Ⅎ𝑎 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
139	134, 138	nfan 1906	. . . 4 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))))
140		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
141		simpll 772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝜑)
142	140	elin1d 4133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ 𝒫 𝐴)
143		elpwi 4536	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
144	142, 143	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ⊆ 𝐴)
145	144	sselda 3915	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝑗 ∈ 𝐴)
146	141, 145, 60	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → 𝐵 ∈ 𝑊)
147	141	adantrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝜑)
148	145	adantrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑗 ∈ 𝐴)
149		simprr 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝑘 ∈ 𝐵)
150	147, 148, 149, 37	syl12anc 842	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
151	140	elin2d 4134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ Fin)
152	29, 32, 140, 146, 150, 151	esum2dlem 34276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑗 ∈ 𝑎Σ^𝑘 ∈ 𝐵𝐶 = Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹)
153		nfv 1921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
154		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑎
155	37	anassrs 468	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
156	155	ralrimiva 3131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
157		nfcv 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐵
158	157	esumcl 34214	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
159	60, 156, 158	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
160	141, 145, 159	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑗 ∈ 𝑎) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
161	153, 154, 151, 160	esumgsum 34229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑗 ∈ 𝑎Σ^𝑘 ∈ 𝐵𝐶 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
162	152, 161	eqtr3d 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^*𝑘 ∈ 𝐵𝐶)))
163		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
164	65	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
165	46	adantlr 721	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
166		iunss1 4936	. . . . . . . . . . . 12 ⊢ (𝑎 ⊆ 𝐴 → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
167	144, 166	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
168	163, 164, 165, 167	esummono 34238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ≤ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
169	162, 168	eqbrtrrd 5096	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ≤ Σ^*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
170	11	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
171	160	ralrimiva 3131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑗 ∈ 𝑎 Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
172	10, 170, 151, 171	gsummptcl 19933	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ (0[,]+∞))
173	9, 172	sselid 3913	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ ℝ^*)
174	70	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 ∈ ℝ^)
175		xrlenlt 11201	. . . . . . . . . 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))))
176	173, 174, 175	syl2anc 590	. . . . . . . . 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))))
177	169, 176	mpbid 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^*𝑘 ∈ 𝐵𝐶)))
178		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝜑
179		eqidd 2740	. . . . . . . . . . 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)))
180	178, 67, 65, 46, 179	esumval 34230	. . . . . . . . . 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^*, < ))
181	180	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → Σ^𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹 = sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^*, < ))
182	181	breq1d 5082	. . . . . . . 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))))
183	177, 182	mtbid 325	. . . . . . 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)))
184	183	adantlr 721	. . . . . 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)))
185	184	adantr 481	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → ¬ sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ) < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
186		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → 𝑠 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
187	186	breq2d 5084	. . . . . 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))))
188	187	notbid 319	. . . . 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))))
189	185, 188	mpbird 258	. . . 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)
190		eqid 2739	. . . . . 6 ⊢ (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)))
191		ovex 7389	. . . . . 6 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V
192	190, 191	elrnmpti 5904	. . . . 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)))
193	192	bilani 505	. . . 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)))
194	139, 189, 193	r19.29af 3248	. . 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)
195	4	nfel1 2917	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 ∈ ℝ^*
196		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑐 <
197		nfcv 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑐ℝ^*
198	6, 197, 196	nfsup 9354	. . . . . . . . . 10 ⊢ Ⅎ𝑐sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )
199	4, 196, 198	nfbr 5119	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < )
200	195, 199	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑐(𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))
201	3, 200	nfan 1906	. . . . . . 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^, < )))
202		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑐𝑢
203	202, 6	nfel 2915	. . . . . . 7 ⊢ Ⅎ𝑐 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
204	201, 203	nfan 1906	. . . . . 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))))
205		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑐 𝑠 < 𝑢
206	204, 205	nfan 1906	. . . . 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)
207		simp-5l 790	. . . . . . . 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)
208		simpr1l 1237	. . . . . . . . . 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^*)
209	208	3anassrs 1367	. . . . . . . . 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^*)
210	209	3anassrs 1367	. . . . . . . 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^*)
211	207, 210	jca 516	. . . . . . 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^*))
212		simpr1r 1238	. . . . . . . . 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^, < ))
213	212	3anassrs 1367	. . . . . . . 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^, < ))
214	213	3anassrs 1367	. . . . . . 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^, < ))
215	211, 214	jca 516	. . . . . 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^*, < )))
216		simpllr 781	. . . . . . 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)
217		simpr 485	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) ∧ 𝑢 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))) ∧ 𝑠 < 𝑢) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑢 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) → 𝑢 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
218	216, 217	breqtrd 5098	. . . . . 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)))
219		simplr 774	. . . . . 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))
220		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
221	220	elin1d 4133	. . . . . . . . . . 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))
222		elpwi 4536	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
223		dmss 5844	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
224		dmiun 5855	. . . . . . . . . . . . . 14 ⊢ dom ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵)
225	223, 224	sseqtrdi 3955	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵))
226		dmxpss 6122	. . . . . . . . . . . . . . . . 17 ⊢ dom ({𝑗} × 𝐵) ⊆ {𝑗}
227	226	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ {𝑗})
228		snssi 4717	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → {𝑗} ⊆ 𝐴)
229	227, 228	sstrd 3925	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐴 → dom ({𝑗} × 𝐵) ⊆ 𝐴)
230	229	rgen 3055	. . . . . . . . . . . . . 14 ⊢ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
231		iunss 4974	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴 ↔ ∀𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴)
232	230, 231	mpbir 232	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ 𝐴 dom ({𝑗} × 𝐵) ⊆ 𝐴
233	225, 232	sstrdi 3927	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ⊆ 𝐴)
234	18	dmex 7849	. . . . . . . . . . . . 13 ⊢ dom 𝑐 ∈ V
235	234	elpw 4533	. . . . . . . . . . . 12 ⊢ (dom 𝑐 ∈ 𝒫 𝐴 ↔ dom 𝑐 ⊆ 𝐴)
236	233, 235	sylibr 235	. . . . . . . . . . 11 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → dom 𝑐 ∈ 𝒫 𝐴)
237	221, 222, 236	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)
238	220	elin2d 4134	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ∈ Fin)
239		dmfi 9235	. . . . . . . . . . 11 ⊢ (𝑐 ∈ Fin → dom 𝑐 ∈ Fin)
240	238, 239	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)
241	237, 240	elind 4129	. . . . . . . . 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))
242		ovex 7389	. . . . . . . . . 10 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)) ∈ V
243	242	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)
244		mpteq1 5161	. . . . . . . . . . 11 ⊢ (𝑎 = dom 𝑐 → (𝑗 ∈ 𝑎 ↦ Σ^𝑘 ∈ 𝐵𝐶) = (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))
245	244	oveq2d 7372	. . . . . . . . . 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)))
246	190, 245	elrnmpt1s 5901	. . . . . . . . 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))))
247	241, 243, 246	syl2anc 590	. . . . . . . 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))))
248		simpr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑡 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶))) → 𝑡 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑗 ∈ dom 𝑐 ↦ Σ^𝑘 ∈ 𝐵𝐶)))
249	248	breq2d 5084	. . . . . . . 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))))
250		simpllr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑠 ∈ ℝ^*)
251	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)
252		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(ℝ^*_𝑠 ↾_s (0[,]+∞))
253		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧 Σ_g
254		nfmpt1 5171	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑐 ↦ 𝐹)
255	252, 253, 254	nfov 7386	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
256	108, 109, 255	nfbr 5119	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 𝑠 < ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))
257	107, 256	nfan 1906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
258		nfv 1921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
259	257, 258	nfan 1906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
260		simp-4l 788	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑧 ∈ 𝑐) → 𝜑)
261	221, 222	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))
262	261	sselda 3915	. . . . . . . . . . . . . 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))
263	260, 262, 46	syl2anc 590	. . . . . . . . . . . . 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[,]+∞))
264	263	ex 413	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (𝑧 ∈ 𝑐 → 𝐹 ∈ (0[,]+∞)))
265	259, 264	ralrimi 3237	. . . . . . . . . . 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[,]+∞))
266	10, 251, 238, 265	gsummptcl 19933	. . . . . . . . . 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[,]+∞))
267	9, 266	sselid 3913	. . . . . . . . 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^)
268		nfv 1921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))
269		nfcv 2901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑐
270	25	nfpw 4548	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
271		nfcv 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗Fin
272	270, 271	nfin 4153	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
273	269, 272	nfel 2915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)
274	268, 273	nfan 1906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
275		simpll 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝜑)
276	78, 233	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ⊆ 𝐴)
277	276	sselda 3915	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑗 ∈ 𝐴)
278	275, 277, 159	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
279	278	adantllr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
280	279	adantllr 725	. . . . . . . . . . . . 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[,]+∞))
281	280	ex 413	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ^) ∧ 𝑠 < ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))) ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → (𝑗 ∈ dom 𝑐 → Σ^*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞)))
282	274, 281	ralrimi 3237	. . . . . . . . . . 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[,]+∞))
283	10, 251, 240, 282	gsummptcl 19933	. . . . . . . . . 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[,]+∞))
284	9, 283	sselid 3913	. . . . . . . . 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^*)
285		simplr 774	. . . . . . . . 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)))
286	23, 273	nfan 1906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin))
287		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
288		xpss 5634	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑗} × 𝐵) ⊆ (V × V)
289	288	rgenw 3057	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V)
290		iunss 4974	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V) ↔ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V))
291	289, 290	mpbir 232	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V)
292	291	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ⊆ (V × V))
293	287, 292	sstrd 3925	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑐 ⊆ (V × V))
294	78, 293	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → 𝑐 ⊆ (V × V))
295		df-rel 5625	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑐 ↔ 𝑐 ⊆ (V × V))
296	294, 295	sylibr 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Rel 𝑐)
297	29, 286, 10, 32, 296, 14, 12, 47	gsummpt2d 33130	. . . . . . . . . . . 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)))))
298		nfcv 2901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗dom 𝑐
299	234	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ V)
300	275	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝜑)
301	277	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝑗 ∈ 𝐴)
302	78	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
303		imass1 6053	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → (𝑐 “ {𝑗}) ⊆ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}))
304	302, 303	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ⊆ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}))
305	58, 60	iunsnima 32710	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}) = 𝐵)
306	275, 277, 305	syl2anc 590	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) “ {𝑗}) = 𝐵)
307	304, 306	sseqtrd 3951	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ⊆ 𝐵)
308	307	sselda 3915	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝑘 ∈ 𝐵)
309	300, 301, 308, 37	syl12anc 842	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ (𝑐 “ {𝑗})) → 𝐶 ∈ (0[,]+∞))
310	309	ralrimiva 3131	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
311		imaexg 7853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ V → (𝑐 “ {𝑗}) ∈ V)
312	18, 311	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 “ {𝑗}) ∈ V
313		nfcv 2901	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑐 “ {𝑗})
314	313	esumcl 34214	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 “ {𝑗}) ∈ V ∧ ∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
315	312, 314	mpan 696	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
316	310, 315	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^*𝑘 ∈ (𝑐 “ {𝑗})𝐶 ∈ (0[,]+∞))
317		nfv 1921	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐)
318	275, 277, 60	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝐵 ∈ 𝑊)
319	275	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝜑)
320	277	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝑗 ∈ 𝐴)
321		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
322	319, 320, 321, 37	syl12anc 842	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
323	317, 318, 322, 307	esummono 34238	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶 ≤ Σ^𝑘 ∈ 𝐵𝐶)
324	286, 298, 299, 316, 278, 323	esumlef 34246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → Σ^𝑗 ∈ dom 𝑐Σ^𝑘 ∈ (𝑐 “ {𝑗})𝐶 ≤ Σ^𝑗 ∈ dom 𝑐Σ^𝑘 ∈ 𝐵𝐶)
325	14, 239	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) → dom 𝑐 ∈ Fin)
326	286, 298, 325, 316	esumgsum 34229	. . . . . . . . . . . . . 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)))
327	14	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → 𝑐 ∈ Fin)
328		imafi2 9261	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ Fin → (𝑐 “ {𝑗}) ∈ Fin)
329	327, 328	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin)) ∧ 𝑗 ∈ dom 𝑐) → (𝑐 “ {𝑗}) ∈ Fin)
330	317, 313, 329, 309	esumgsum 34229	. . . . . . . . . . . . . . . 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)))
331	286, 330	mpteq2da 5164	. . . . . . . . . . . . . . 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))))
332	331	oveq2d 7372	. . . . . . . . . . . . . 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)))))
333	326, 332	eqtrd 2774	. . . . . . . . . . . . 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)))))
334	286, 298, 325, 278	esumgsum 34229	. . . . . . . . . . . . 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)))
335	324, 333, 334	3brtr3d 5103	. . . . . . . . . . . 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)))
336	297, 335	eqbrtrd 5094	. . . . . . . . . . 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)))
337	336	adantlr 721	. . . . . . . . . 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)))
338	337	adantlr 721	. . . . . . . . 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)))
339	250, 267, 284, 285, 338	xrltletrd 13103	. . . . . . . 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)))
340	247, 249, 339	rspcedvd 3562	. . . . . . 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)
341	340	adantllr 725	. . . . . 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)
342	215, 218, 219, 341	syl21anc 843	. . . . 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)
343	52, 87	elrnmpti 5904	. . . . . . 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)))
344	343	biimpi 217	. . . . . 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)))
345	344	ad2antlr 733	. . . . 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)))
346	206, 342, 345	r19.29af 3248	. . . 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)
347	2, 132	suplub 9363	. . . . . 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))
348	347	imp 407	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ^* ∧ 𝑠 < sup(ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹))), ℝ^, < ))) → ∃𝑡 ∈ ran (𝑐 ∈ (𝒫 ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑧 ∈ 𝑐 ↦ 𝐹)))𝑠 < 𝑡)
349		breq2 5076	. . . . . 6 ⊢ (𝑡 = 𝑢 → (𝑠 < 𝑡 ↔ 𝑠 < 𝑢))
350	349	cbvrexvw 3218	. . . . 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)
351	348, 350	sylib 219	. . . 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)
352	346, 351	r19.29a 3147	. . 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)
353	2, 133, 194, 352	eqsupd 9360	. 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^*, < ))
354		nfcv 2901	. . 3 ⊢ Ⅎ𝑗𝐴
355		eqidd 2740	. . 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)))
356	23, 354, 58, 159, 355	esumval 34230	. 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^*, < ))
357	353, 356, 180	3eqtr4d 2784	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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 Ⅎwnfc 2886 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ∩ cin 3882 ⊆ wss 3883 𝒫 cpw 4529 {csn 4555 ⟨cop 4561 ∪ ciun 4921 class class class wbr 5072 ↦ cmpt 5153 Or wor 5525 × cxp 5616 dom cdm 5618 ran crn 5619 “ cima 5621 Rel wrel 5623 (class class class)co 7356 Fincfn 8883 supcsup 9343 0cc0 11029 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 [,]cicc 13292 ↾_s cress 17191 Σ_g cgsu 17394 ℝ^*_𝑠cxrs 17455 CMndccmn 19746 Σ^*cesum 34211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-ordt 17456 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-ps 18523 df-tsr 18524 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-subrng 20518 df-subrg 20542 df-abv 20781 df-lmod 20852 df-scaf 20853 df-sra 21163 df-rgmod 21164 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-tmd 24055 df-tgp 24056 df-tsms 24110 df-trg 24143 df-xms 24303 df-ms 24304 df-tms 24305 df-nm 24565 df-ngp 24566 df-nrg 24568 df-nlm 24569 df-ii 24862 df-cncf 24863 df-limc 25851 df-dv 25852 df-log 26538 df-esum 34212

This theorem is referenced by: esumiun 34278

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