sge0f1o - Mathbox for Glauco Siliprandi

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Theorem sge0f1o 40599

Description: Re-index a nonnegative extended sum using a bijection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
sge0f1o.1	⊢ Ⅎ𝑘𝜑
sge0f1o.2	⊢ Ⅎ𝑛𝜑
sge0f1o.3	⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
sge0f1o.4	⊢ (𝜑 → 𝐶 ∈ 𝑉)
sge0f1o.5	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
sge0f1o.6	⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
sge0f1o.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))

**Assertion**
Ref	Expression
sge0f1o	⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)))

Distinct variable groups: 𝐴,𝑘,𝑛 𝐵,𝑛 𝐶,𝑘,𝑛 𝐷,𝑘 𝑘,𝐹,𝑛 𝑘,𝐺 Allowed substitution hints: 𝜑(𝑘,𝑛) 𝐵(𝑘) 𝐷(𝑛) 𝐺(𝑛) 𝑉(𝑘,𝑛)

Proof of Theorem sge0f1o Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0f1o.4	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
2		sge0f1o.5	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
3		f1ofo 6144	. . . . . . 7 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–onto→𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)
5		fornex 7135	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → (𝐹:𝐶–onto→𝐴 → 𝐴 ∈ V))
6	1, 4, 5	sylc 65	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐴 ∈ V)
8		sge0f1o.1	. . . . . 6 ⊢ Ⅎ𝑘𝜑
9		sge0f1o.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
10		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
11	8, 9, 10	fmptdf 6387	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
12	11	adantr 481	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
13		pnfex 10093	. . . . . . . 8 ⊢ +∞ ∈ V
14		eqid 2622	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷)
15	14	elrnmpt 5372	. . . . . . . 8 ⊢ (+∞ ∈ V → (+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷) ↔ ∃𝑛 ∈ 𝐶 +∞ = 𝐷))
16	13, 15	ax-mp 5	. . . . . . 7 ⊢ (+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷) ↔ ∃𝑛 ∈ 𝐶 +∞ = 𝐷)
17	16	biimpi 206	. . . . . 6 ⊢ (+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷) → ∃𝑛 ∈ 𝐶 +∞ = 𝐷)
18	17	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → ∃𝑛 ∈ 𝐶 +∞ = 𝐷)
19		sge0f1o.2	. . . . . . 7 ⊢ Ⅎ𝑛𝜑
20		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑛+∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)
21		simp3 1063	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → +∞ = 𝐷)
22		f1of 6137	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
23	2, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
24	23	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
25		sge0f1o.6	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
26		nfcv 2764	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘𝑛)
27		nfv 1843	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐹‘𝑛) = 𝐺
28	26	nfcsb1 3548	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵
29		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝐷
30	28, 29	nfeq 2776	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷
31	27, 30	nfim 1825	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝐹‘𝑛) = 𝐺 → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)
32		eqeq1 2626	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐹‘𝑛) → (𝑘 = 𝐺 ↔ (𝐹‘𝑛) = 𝐺))
33		csbeq1a 3542	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
34	33	eqeq1d 2624	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 = 𝐷 ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷))
35	32, 34	imbi12d 334	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐹‘𝑛) → ((𝑘 = 𝐺 → 𝐵 = 𝐷) ↔ ((𝐹‘𝑛) = 𝐺 → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)))
36		sge0f1o.3	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
37	26, 31, 35, 36	vtoclgf 3264	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ 𝐴 → ((𝐹‘𝑛) = 𝐺 → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷))
38	24, 25, 37	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)
39	38	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
40	39	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → 𝐷 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
41	21, 40	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → +∞ = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
42		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝜑)
43	42, 24	jca 554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴))
44		nfv 1843	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐹‘𝑛) ∈ 𝐴
45	8, 44	nfan 1828	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴)
46	28	nfel1 2779	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)
47	45, 46	nfim 1825	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞))
48		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐹‘𝑛) → (𝑘 ∈ 𝐴 ↔ (𝐹‘𝑛) ∈ 𝐴))
49	48	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐹‘𝑛) → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴)))
50	33	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 ∈ (0[,]+∞) ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)))
51	49, 50	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐹‘𝑛) → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞))))
52	26, 47, 51, 9	vtoclgf 3264	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ 𝐴 → ((𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)))
53	24, 43, 52	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞))
54	28, 10, 33	elrnmpt1sf 39376	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑛) ∈ 𝐴 ∧ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
55	24, 53, 54	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
56	55	3adant3 1081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
57	41, 56	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
58	57	3exp 1264	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐶 → (+∞ = 𝐷 → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))))
59	19, 20, 58	rexlimd 3026	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ 𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
60	59	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (∃𝑛 ∈ 𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
61	18, 60	mpd 15	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
62	7, 12, 61	sge0pnfval 40590	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)
63	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐶 ∈ 𝑉)
64	39, 53	eqeltrd 2701	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ (0[,]+∞))
65	19, 64, 14	fmptdf 6387	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐶 ↦ 𝐷):𝐶⟶(0[,]+∞))
66	65	adantr 481	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (𝑛 ∈ 𝐶 ↦ 𝐷):𝐶⟶(0[,]+∞))
67		simpr 477	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
68	63, 66, 67	sge0pnfval 40590	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)) = +∞)
69	62, 68	eqtr4d 2659	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
70		sumex 14418	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝑦 𝐵 ∈ V
71	70	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘 ∈ 𝑦 𝐵 ∈ V)
72		cnvimass 5485	. . . . . . . . . . . . 13 ⊢ (^◡𝐹 “ 𝑦) ⊆ dom 𝐹
73	72	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝐹 “ 𝑦) ⊆ dom 𝐹)
74		fdm 6051	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐶⟶𝐴 → dom 𝐹 = 𝐶)
75	23, 74	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = 𝐶)
76	73, 75	sseqtrd 3641	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 “ 𝑦) ⊆ 𝐶)
77		fex 6490	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐹 ∈ V)
78	23, 1, 77	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
79		cnvexg 7112	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
80	78, 79	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ^◡𝐹 ∈ V)
81		imaexg 7103	. . . . . . . . . . . . 13 ⊢ (^◡𝐹 ∈ V → (^◡𝐹 “ 𝑦) ∈ V)
82	80, 81	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝐹 “ 𝑦) ∈ V)
83		elpwg 4166	. . . . . . . . . . . 12 ⊢ ((^◡𝐹 “ 𝑦) ∈ V → ((^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶 ↔ (^◡𝐹 “ 𝑦) ⊆ 𝐶))
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶 ↔ (^◡𝐹 “ 𝑦) ⊆ 𝐶))
85	76, 84	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶)
86	85	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶)
87		f1ocnv 6149	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐶)
88	2, 87	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝐹:𝐴–1-1-onto→𝐶)
89		f1ofun 6139	. . . . . . . . . . . 12 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐶 → Fun ^◡𝐹)
90	88, 89	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun ^◡𝐹)
91	90	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Fun ^◡𝐹)
92		elinel2 3800	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
93	92	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
94		imafi 8259	. . . . . . . . . 10 ⊢ ((Fun ^◡𝐹 ∧ 𝑦 ∈ Fin) → (^◡𝐹 “ 𝑦) ∈ Fin)
95	91, 93, 94	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ Fin)
96	86, 95	elind 3798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
97	96	adantlr 751	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
98		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑘 ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)
99	8, 98	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
100		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
101	99, 100	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
102		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛+∞
103		nfmpt1 4747	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑛 ∈ 𝐶 ↦ 𝐷)
104	103	nfrn 5368	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛ran (𝑛 ∈ 𝐶 ↦ 𝐷)
105	102, 104	nfel 2777	. . . . . . . . . . 11 ⊢ Ⅎ𝑛+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)
106	105	nfn 1784	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)
107	19, 106	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
108		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑛 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
109	107, 108	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
110	95	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ Fin)
111		f1of1 6136	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–1-1→𝐴)
112	2, 111	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐴)
113	112	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶–1-1→𝐴)
114	84	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶 ↔ (^◡𝐹 “ 𝑦) ⊆ 𝐶))
115	86, 114	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ⊆ 𝐶)
116		f1ores 6151	. . . . . . . . . . 11 ⊢ ((𝐹:𝐶–1-1→𝐴 ∧ (^◡𝐹 “ 𝑦) ⊆ 𝐶) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→(𝐹 “ (^◡𝐹 “ 𝑦)))
117	113, 115, 116	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→(𝐹 “ (^◡𝐹 “ 𝑦)))
118	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶–onto→𝐴)
119		elpwinss 39216	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
120	119	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
121		foimacnv 6154	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶–onto→𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
122	118, 120, 121	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
123	122	f1oeq3d 6134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→(𝐹 “ (^◡𝐹 “ 𝑦)) ↔ (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→𝑦))
124	117, 123	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→𝑦)
125	124	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→𝑦)
126	82	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → (^◡𝐹 “ 𝑦) ∈ V)
127		simpll 790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → 𝜑)
128	96	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
129		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → 𝑛 ∈ (^◡𝐹 “ 𝑦))
130	127, 128, 129	jca31 557	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)))
131		eleq1 2689	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)))
132	131	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ (𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
133		eleq2 2690	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝑛 ∈ 𝑥 ↔ 𝑛 ∈ (^◡𝐹 “ 𝑦)))
134	132, 133	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) ↔ ((𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦))))
135		reseq2 5391	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (^◡𝐹 “ 𝑦)))
136	135	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → ((𝐹 ↾ 𝑥)‘𝑛) = ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛))
137	136	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (((𝐹 ↾ 𝑥)‘𝑛) = 𝐺 ↔ ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺))
138	134, 137	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑛) = 𝐺) ↔ (((𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺)))
139		fvres 6207	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑥 → ((𝐹 ↾ 𝑥)‘𝑛) = (𝐹‘𝑛))
140	139	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑛) = (𝐹‘𝑛))
141		simpll 790	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → 𝜑)
142		elpwinss 39216	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ⊆ 𝐶)
143	142	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ⊆ 𝐶)
144	143	sselda 3603	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐶)
145	141, 144, 25	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → (𝐹‘𝑛) = 𝐺)
146	140, 145	eqtrd 2656	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑛) = 𝐺)
147	138, 146	vtoclg 3266	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑦) ∈ V → (((𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺))
148	126, 130, 147	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺)
149	148	adantllr 755	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺)
150	82	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ V)
151		simpll 790	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)))
152	85	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶)
153	110	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ Fin)
154	152, 153	elind 3798	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
155		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝑦)
156	122	eqcomd 2628	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 = (𝐹 “ (^◡𝐹 “ 𝑦)))
157	156	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑦 = (𝐹 “ (^◡𝐹 “ 𝑦)))
158	155, 157	eleqtrd 2703	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦)))
159	158	adantllr 755	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦)))
160	151, 154, 159	jca31 557	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦))))
161	131	anbi2d 740	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
162		imaeq2 5462	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (^◡𝐹 “ 𝑦)))
163	162	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝑘 ∈ (𝐹 “ 𝑥) ↔ 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦))))
164	161, 163	anbi12d 747	. . . . . . . . . . 11 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) ↔ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦)))))
165	164	imbi1d 331	. . . . . . . . . 10 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝐵 ∈ ℂ) ↔ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦))) → 𝐵 ∈ ℂ)))
166		rge0ssre 12280	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
167		ax-resscn 9993	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
168	166, 167	sstri 3612	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℂ
169		simplll 798	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝜑)
170		simpllr 799	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
171		fimass 6081	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐶⟶𝐴 → (𝐹 “ 𝑥) ⊆ 𝐴)
172	23, 171	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 “ 𝑥) ⊆ 𝐴)
173	172	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → (𝐹 “ 𝑥) ⊆ 𝐴)
174		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝑘 ∈ (𝐹 “ 𝑥))
175	173, 174	sseldd 3604	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝑘 ∈ 𝐴)
176	175	adantllr 755	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝑘 ∈ 𝐴)
177		foelrni 6244	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐶–onto→𝐴 ∧ 𝑘 ∈ 𝐴) → ∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘)
178	4, 177	sylan 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘)
179	178	adantlr 751	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → ∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘)
180		nfv 1843	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛 𝑘 ∈ 𝐴
181	107, 180	nfan 1828	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴)
182		nfv 1843	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 𝐵 ∈ (0[,)+∞)
183		csbid 3541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
184	183	eqcomi 2631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐵 = ⦋𝑘 / 𝑘⦌𝐵
185	184	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 = ⦋𝑘 / 𝑘⦌𝐵)
186		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑛) = 𝑘 → (𝐹‘𝑛) = 𝑘)
187	186	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑛) = 𝑘 → 𝑘 = (𝐹‘𝑛))
188	187	csbeq1d 3540	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑛) = 𝑘 → ⦋𝑘 / 𝑘⦌𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
189	188	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → ⦋𝑘 / 𝑘⦌𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
190	38	idi 2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)
191	190	3adant3 1081	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)
192	185, 189, 191	3eqtrd 2660	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 = 𝐷)
193	192	3adant1r 1319	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 = 𝐷)
194		0xr 10086	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℝ^*
195	194	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 0 ∈ ℝ^*)
196		pnfxr 10092	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ +∞ ∈ ℝ^*
197	196	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ℝ^*)
198	64	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ (0[,]+∞))
199		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ 𝐷 ∈ (0[,)+∞))
200	195, 197, 198, 199	eliccnelico 39756	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 = +∞)
201	200	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ = 𝐷)
202		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝑛 ∈ 𝐶)
203	64	idi 2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ (0[,]+∞))
204	14	elrnmpt1 5374	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝐷 ∈ (0[,]+∞)) → 𝐷 ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
205	202, 203, 204	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
206	205	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
207	201, 206	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
208	207	adantllr 755	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
209		simpllr 799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
210	208, 209	condan 835	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ (0[,)+∞))
211	210	3adant3 1081	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐷 ∈ (0[,)+∞))
212	193, 211	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 ∈ (0[,)+∞))
213	212	3exp 1264	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (𝑛 ∈ 𝐶 → ((𝐹‘𝑛) = 𝑘 → 𝐵 ∈ (0[,)+∞))))
214	213	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝐶 → ((𝐹‘𝑛) = 𝑘 → 𝐵 ∈ (0[,)+∞))))
215	181, 182, 214	rexlimd 3026	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → (∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘 → 𝐵 ∈ (0[,)+∞)))
216	179, 215	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
217	169, 170, 176, 216	syl21anc 1325	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝐵 ∈ (0[,)+∞))
218	168, 217	sseldi 3601	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝐵 ∈ ℂ)
219	218	idi 2	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝐵 ∈ ℂ)
220	165, 219	vtoclg 3266	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑦) ∈ V → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦))) → 𝐵 ∈ ℂ))
221	150, 160, 220	sylc 65	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
222	101, 109, 36, 110, 125, 149, 221	fsumf1of 39806	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ (^◡𝐹 “ 𝑦)𝐷)
223		sumeq1 14419	. . . . . . . . 9 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑛 ∈ (^◡𝐹 “ 𝑦)𝐷)
224	223	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ 𝑥 𝐷 ↔ Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ (^◡𝐹 “ 𝑦)𝐷))
225	224	rspcev 3309	. . . . . . 7 ⊢ (((^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ (^◡𝐹 “ 𝑦)𝐷) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ 𝑥 𝐷)
226	97, 222, 225	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ 𝑥 𝐷)
227	71, 226	rnmptssrn 39368	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷))
228		sumex 14418	. . . . . . 7 ⊢ Σ𝑛 ∈ 𝑥 𝐷 ∈ V
229	228	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛 ∈ 𝑥 𝐷 ∈ V)
230	6, 172	ssexd 4805	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 “ 𝑥) ∈ V)
231		elpwg 4166	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝑥) ∈ V → ((𝐹 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
232	230, 231	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐹 “ 𝑥) ⊆ 𝐴))
233	172, 232	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 “ 𝑥) ∈ 𝒫 𝐴)
234	233	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ 𝒫 𝐴)
235		ffun 6048	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶⟶𝐴 → Fun 𝐹)
236	23, 235	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
237	236	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Fun 𝐹)
238		elinel2 3800	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ∈ Fin)
239	238	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
240		imafi 8259	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ Fin) → (𝐹 “ 𝑥) ∈ Fin)
241	237, 239, 240	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ Fin)
242	234, 241	elind 3798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ (𝒫 𝐴 ∩ Fin))
243	242	adantlr 751	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ (𝒫 𝐴 ∩ Fin))
244		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
245	99, 244	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
246		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
247	107, 246	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
248	238	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
249	112	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐶–1-1→𝐴)
250		f1ores 6151	. . . . . . . . . . 11 ⊢ ((𝐹:𝐶–1-1→𝐴 ∧ 𝑥 ⊆ 𝐶) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→(𝐹 “ 𝑥))
251	249, 143, 250	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→(𝐹 “ 𝑥))
252	251	adantlr 751	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→(𝐹 “ 𝑥))
253	146	adantllr 755	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑛) = 𝐺)
254	245, 247, 36, 248, 252, 253, 218	fsumf1of 39806	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵 = Σ𝑛 ∈ 𝑥 𝐷)
255	254	eqcomd 2628	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵)
256		sumeq1 14419	. . . . . . . . 9 ⊢ (𝑦 = (𝐹 “ 𝑥) → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵)
257	256	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑦 = (𝐹 “ 𝑥) → (Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ 𝑦 𝐵 ↔ Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵))
258	257	rspcev 3309	. . . . . . 7 ⊢ (((𝐹 “ 𝑥) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ 𝑦 𝐵)
259	243, 255, 258	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ 𝑦 𝐵)
260	229, 259	rnmptssrn 39368	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷) ⊆ ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵))
261	227, 260	eqssd 3620	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵) = ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷))
262	261	supeq1d 8352	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵), ℝ^, < ) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷), ℝ^, < ))
263	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐴 ∈ V)
264	99, 263, 216	sge0revalmpt 40595	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵), ℝ^*, < ))
265	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐶 ∈ 𝑉)
266	107, 265, 210	sge0revalmpt 40595	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷), ℝ^*, < ))
267	262, 264, 266	3eqtr4d 2666	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
268	69, 267	pm2.61dan 832	1 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ⦋csb 3533 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ↦ cmpt 4729 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 [,)cico 12177 [,]cicc 12178 Σcsu 14416 Σ^{^}csumge0 40579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-sumge0 40580

This theorem is referenced by: sge0resrnlem 40620 sge0fodjrnlem 40633 sge0xp 40646 meadjiunlem 40682 isomenndlem 40744 ovnsubaddlem1 40784

W3C validator