sge0f1o - Mathbox for Glauco Siliprandi

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Theorem sge0f1o 47154

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 6832	. . . . . . 7 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–onto→𝐴)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)
5		focdmex 7959	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → (𝐹:𝐶–onto→𝐴 → 𝐴 ∈ V))
6	1, 4, 5	sylc 66	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐴 ∈ V)
8		sge0f1o.1	. . . . . 6 ⊢ Ⅎ𝑘𝜑
9		sge0f1o.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
10	8, 9	fmptd2f 46008	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
12		sge0f1o.2	. . . . . 6 ⊢ Ⅎ𝑛𝜑
13		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑛+∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)
14		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → +∞ = 𝐷)
15		f1of 6824	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
16	2, 15	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
17	16	ffvelcdmda 7083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
18		sge0f1o.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
19		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘𝑛) = 𝐺
20		nfcsb1v 3878	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵
21	20	nfeq1 2942	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷
22	19, 21	nfim 1929	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝐹‘𝑛) = 𝐺 → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)
23		eqeq1 2769	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐹‘𝑛) → (𝑘 = 𝐺 ↔ (𝐹‘𝑛) = 𝐺))
24		csbeq1a 3868	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
25	24	eqeq1d 2767	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 = 𝐷 ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷))
26	23, 25	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐹‘𝑛) → ((𝑘 = 𝐺 → 𝐵 = 𝐷) ↔ ((𝐹‘𝑛) = 𝐺 → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)))
27		sge0f1o.3	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
28	22, 26, 27	vtoclg1f 3537	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ 𝐴 → ((𝐹‘𝑛) = 𝐺 → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷))
29	17, 18, 28	sylc 66	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)
30	29	eqcomd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
31	30	3adant3 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → 𝐷 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
32	14, 31	eqtrd 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → +∞ = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
33		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝜑)
34	33, 17	jca 521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴))
35		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘𝑛) ∈ 𝐴
36	8, 35	nfan 1932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴)
37	20	nfel1 2943	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)
38	36, 37	nfim 1929	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞))
39		eleq1 2853	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐹‘𝑛) → (𝑘 ∈ 𝐴 ↔ (𝐹‘𝑛) ∈ 𝐴))
40	39	anbi2d 642	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐹‘𝑛) → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴)))
41	24	eleq1d 2850	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 ∈ (0[,]+∞) ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)))
42	40, 41	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝐹‘𝑛) → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞))))
43	38, 42, 9	vtoclg1f 3537	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ 𝐴 → ((𝜑 ∧ (𝐹‘𝑛) ∈ 𝐴) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)))
44	17, 34, 43	sylc 66	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞))
45		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
46	20, 45, 24	elrnmpt1sf 45965	. . . . . . . . . 10 ⊢ (((𝐹‘𝑛) ∈ 𝐴 ∧ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ (0[,]+∞)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
47	17, 44, 46	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
48	47	3adant3 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
49	32, 48	eqeltrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ +∞ = 𝐷) → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
50	49	3exp 1137	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝐶 → (+∞ = 𝐷 → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))))
51	12, 13, 50	rexlimd 3274	. . . . 5 ⊢ (𝜑 → (∃𝑛 ∈ 𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
52		pnfex 11277	. . . . . . 7 ⊢ +∞ ∈ V
53		eqid 2765	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷)
54	53	elrnmpt 5950	. . . . . . 7 ⊢ (+∞ ∈ V → (+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷) ↔ ∃𝑛 ∈ 𝐶 +∞ = 𝐷))
55	52, 54	ax-mp 5	. . . . . 6 ⊢ (+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷) ↔ ∃𝑛 ∈ 𝐶 +∞ = 𝐷)
56	55	biimpi 219	. . . . 5 ⊢ (+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷) → ∃𝑛 ∈ 𝐶 +∞ = 𝐷)
57	51, 56	impel 515	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
58	7, 11, 57	sge0pnfval 47145	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)
59	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐶 ∈ 𝑉)
60	30, 44	eqeltrd 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ (0[,]+∞))
61	12, 60	fmptd2f 46008	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐶 ↦ 𝐷):𝐶⟶(0[,]+∞))
62	61	adantr 486	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (𝑛 ∈ 𝐶 ↦ 𝐷):𝐶⟶(0[,]+∞))
63		simpr 490	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
64	59, 62, 63	sge0pnfval 47145	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)) = +∞)
65	58, 64	eqtr4d 2803	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
66		sumex 15763	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝑦 𝐵 ∈ V
67	66	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘 ∈ 𝑦 𝐵 ∈ V)
68		cnvimass 6086	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ 𝑦) ⊆ dom 𝐹
69	68, 16	fssdm 6729	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 “ 𝑦) ⊆ 𝐶)
70	1, 69	sselpwd 5301	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶)
71	70	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶)
72		f1ocnv 6837	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐶)
73	2, 72	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐹:𝐴–1-1-onto→𝐶)
74		f1ofun 6826	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐶 → Fun ^◡𝐹)
75	73, 74	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → Fun ^◡𝐹)
76		elinel2 4155	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
77		imafi 9282	. . . . . . . . . 10 ⊢ ((Fun ^◡𝐹 ∧ 𝑦 ∈ Fin) → (^◡𝐹 “ 𝑦) ∈ Fin)
78	75, 76, 77	syl2an 608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ Fin)
79	71, 78	elind 4153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
80	79	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
81		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑘 ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)
82	8, 81	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
83		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
84	82, 83	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
85		nfmpt1 5212	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑛 ∈ 𝐶 ↦ 𝐷)
86	85	nfrn 5944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛ran (𝑛 ∈ 𝐶 ↦ 𝐷)
87	86	nfel2 2945	. . . . . . . . . . 11 ⊢ Ⅎ𝑛+∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)
88	87	nfn 1890	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)
89	12, 88	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
90		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑛 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
91	89, 90	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
92	78	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ∈ Fin)
93		f1of1 6823	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–1-1→𝐴)
94	2, 93	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐴)
95	94	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶–1-1→𝐴)
96	69	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (^◡𝐹 “ 𝑦) ⊆ 𝐶)
97		f1ores 6839	. . . . . . . . . . 11 ⊢ ((𝐹:𝐶–1-1→𝐴 ∧ (^◡𝐹 “ 𝑦) ⊆ 𝐶) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→(𝐹 “ (^◡𝐹 “ 𝑦)))
98	95, 96, 97	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→(𝐹 “ (^◡𝐹 “ 𝑦)))
99		elpwinss 45827	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
100		foimacnv 6842	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶–onto→𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
101	4, 99, 100	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
102	101	f1oeq3d 6821	. . . . . . . . . 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))
103	98, 102	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→𝑦)
104	103	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (^◡𝐹 “ 𝑦)):(^◡𝐹 “ 𝑦)–1-1-onto→𝑦)
105	16, 1	fexd 7232	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
106		cnvexg 7927	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
107	105, 106	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝐹 ∈ V)
108	107	imaexd 7919	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 “ 𝑦) ∈ V)
109	108	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → (^◡𝐹 “ 𝑦) ∈ V)
110		simpll 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → 𝜑)
111	79	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
112		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → 𝑛 ∈ (^◡𝐹 “ 𝑦))
113	110, 111, 112	jca31 524	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)))
114		eleq1 2853	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)))
115	114	anbi2d 642	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ (𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
116		eleq2w2 2761	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝑛 ∈ 𝑥 ↔ 𝑛 ∈ (^◡𝐹 “ 𝑦)))
117	115, 116	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) ↔ ((𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦))))
118		reseq2 5975	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (^◡𝐹 “ 𝑦)))
119	118	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → ((𝐹 ↾ 𝑥)‘𝑛) = ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛))
120	119	eqeq1d 2767	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (((𝐹 ↾ 𝑥)‘𝑛) = 𝐺 ↔ ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺))
121	117, 120	imbi12d 347	. . . . . . . . . . 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)))
122		fvres 6904	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑥 → ((𝐹 ↾ 𝑥)‘𝑛) = (𝐹‘𝑛))
123	122	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑛) = (𝐹‘𝑛))
124		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → 𝜑)
125		elpwinss 45827	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ⊆ 𝐶)
126	125	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ⊆ 𝐶)
127	126	sselda 3938	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ 𝐶)
128	124, 127, 18	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → (𝐹‘𝑛) = 𝐺)
129	123, 128	eqtrd 2800	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑛) = 𝐺)
130	121, 129	vtoclg 3524	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑦) ∈ V → (((𝜑 ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺))
131	109, 113, 130	sylc 66	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺)
132	131	adantllr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (^◡𝐹 “ 𝑦)) → ((𝐹 ↾ (^◡𝐹 “ 𝑦))‘𝑛) = 𝐺)
133	108	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ V)
134		simpll 779	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)))
135	70	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ 𝒫 𝐶)
136	92	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ Fin)
137	135, 136	elind 4153	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
138		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝑦)
139	101	eqcomd 2771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 = (𝐹 “ (^◡𝐹 “ 𝑦)))
140	139	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑦 = (𝐹 “ (^◡𝐹 “ 𝑦)))
141	138, 140	eleqtrd 2867	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦)))
142	141	adantllr 732	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦)))
143	134, 137, 142	jca31 524	. . . . . . . . 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))))
144	114	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
145		imaeq2 6060	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (^◡𝐹 “ 𝑦)))
146	145	eleq2d 2851	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → (𝑘 ∈ (𝐹 “ 𝑥) ↔ 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦))))
147	144, 146	anbi12d 644	. . . . . . . . . . 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)))))
148	147	imbi1d 344	. . . . . . . . . 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)))
149		rge0ssre 13499	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
150		ax-resscn 11172	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
151	149, 150	sstri 3947	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℂ
152		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝜑)
153		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
154	16	fimassd 6731	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 “ 𝑥) ⊆ 𝐴)
155	154	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → (𝐹 “ 𝑥) ⊆ 𝐴)
156		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝑘 ∈ (𝐹 “ 𝑥))
157	155, 156	sseldd 3939	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝑘 ∈ 𝐴)
158	157	adantllr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝑘 ∈ 𝐴)
159		foelcdmi 6946	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐶–onto→𝐴 ∧ 𝑘 ∈ 𝐴) → ∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘)
160	4, 159	sylan 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘)
161	160	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → ∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘)
162		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 𝑘 ∈ 𝐴
163	89, 162	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴)
164		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝐵 ∈ (0[,)+∞)
165		csbid 3867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
166	165	eqcomi 2774	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = ⦋𝑘 / 𝑘⦌𝐵
167	166	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 = ⦋𝑘 / 𝑘⦌𝐵)
168		id 23	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑛) = 𝑘 → (𝐹‘𝑛) = 𝑘)
169	168	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑛) = 𝑘 → 𝑘 = (𝐹‘𝑛))
170	169	csbeq1d 3858	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑛) = 𝑘 → ⦋𝑘 / 𝑘⦌𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
171	170	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → ⦋𝑘 / 𝑘⦌𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
172	29	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐷)
173	167, 171, 172	3eqtrd 2804	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 = 𝐷)
174	173	3adant1r 1196	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 = 𝐷)
175		0xr 11271	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℝ^*
176	175	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 0 ∈ ℝ^*)
177		pnfxr 11278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ +∞ ∈ ℝ^*
178	177	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ℝ^*)
179	60	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ (0[,]+∞))
180		simpr 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ 𝐷 ∈ (0[,)+∞))
181	176, 178, 179, 180	eliccnelico 46303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 = +∞)
182	181	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ = 𝐷)
183		simpr 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝑛 ∈ 𝐶)
184	53, 183, 60	elrnmpt1d 5956	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
185	184	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
186	182, 185	eqeltrd 2865	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
187	186	adantllr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
188		simpllr 788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷))
189	187, 188	condan 830	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ (0[,)+∞))
190	189	3adant3 1150	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐷 ∈ (0[,)+∞))
191	174, 190	eqeltrd 2865	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) = 𝑘) → 𝐵 ∈ (0[,)+∞))
192	191	3exp 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (𝑛 ∈ 𝐶 → ((𝐹‘𝑛) = 𝑘 → 𝐵 ∈ (0[,)+∞))))
193	192	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝐶 → ((𝐹‘𝑛) = 𝑘 → 𝐵 ∈ (0[,)+∞))))
194	163, 164, 193	rexlimd 3274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → (∃𝑛 ∈ 𝐶 (𝐹‘𝑛) = 𝑘 → 𝐵 ∈ (0[,)+∞)))
195	161, 194	mpd 16	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
196	152, 153, 158, 195	syl21anc 851	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝐵 ∈ (0[,)+∞))
197	151, 196	sselid 3936	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ 𝑥)) → 𝐵 ∈ ℂ)
198	148, 197	vtoclg 3524	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑦) ∈ V → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ (^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (^◡𝐹 “ 𝑦))) → 𝐵 ∈ ℂ))
199	133, 143, 198	sylc 66	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
200	84, 91, 27, 92, 104, 132, 199	fsumf1of 46348	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ (^◡𝐹 “ 𝑦)𝐷)
201		sumeq1 15764	. . . . . . . 8 ⊢ (𝑥 = (^◡𝐹 “ 𝑦) → Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑛 ∈ (^◡𝐹 “ 𝑦)𝐷)
202	201	rspceeqv 3606	. . . . . . 7 ⊢ (((^◡𝐹 “ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ (^◡𝐹 “ 𝑦)𝐷) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ 𝑥 𝐷)
203	80, 200, 202	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑛 ∈ 𝑥 𝐷)
204	67, 203	rnmptssrn 45958	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷))
205		sumex 15763	. . . . . . 7 ⊢ Σ𝑛 ∈ 𝑥 𝐷 ∈ V
206	205	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛 ∈ 𝑥 𝐷 ∈ V)
207	6, 154	sselpwd 5301	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 “ 𝑥) ∈ 𝒫 𝐴)
208	207	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ 𝒫 𝐴)
209	16	ffund 6714	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
210		elinel2 4155	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ∈ Fin)
211		imafi 9282	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ Fin) → (𝐹 “ 𝑥) ∈ Fin)
212	209, 210, 211	syl2an 608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ Fin)
213	208, 212	elind 4153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ (𝒫 𝐴 ∩ Fin))
214	213	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 “ 𝑥) ∈ (𝒫 𝐴 ∩ Fin))
215		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
216	82, 215	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
217		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
218	89, 217	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
219	210	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
220		f1ores 6839	. . . . . . . . . . 11 ⊢ ((𝐹:𝐶–1-1→𝐴 ∧ 𝑥 ⊆ 𝐶) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→(𝐹 “ 𝑥))
221	94, 125, 220	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→(𝐹 “ 𝑥))
222	221	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→(𝐹 “ 𝑥))
223	129	adantllr 732	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑛) = 𝐺)
224	216, 218, 27, 219, 222, 223, 197	fsumf1of 46348	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵 = Σ𝑛 ∈ 𝑥 𝐷)
225	224	eqcomd 2771	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵)
226		sumeq1 15764	. . . . . . . 8 ⊢ (𝑦 = (𝐹 “ 𝑥) → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵)
227	226	rspceeqv 3606	. . . . . . 7 ⊢ (((𝐹 “ 𝑥) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ (𝐹 “ 𝑥)𝐵) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ 𝑦 𝐵)
228	214, 225, 227	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛 ∈ 𝑥 𝐷 = Σ𝑘 ∈ 𝑦 𝐵)
229	206, 228	rnmptssrn 45958	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷) ⊆ ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵))
230	204, 229	eqssd 3955	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵) = ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷))
231	230	supeq1d 9413	. . 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^, < ))
232	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐴 ∈ V)
233	82, 232, 195	sge0revalmpt 47150	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑦 𝐵), ℝ^*, < ))
234	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → 𝐶 ∈ 𝑉)
235	89, 234, 189	sge0revalmpt 47150	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 𝐷), ℝ^*, < ))
236	231, 233, 235	3eqtr4d 2810	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛 ∈ 𝐶 ↦ 𝐷)) → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
237	65, 236	pm2.61dan 825	1 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^{^}‘(𝑛 ∈ 𝐶 ↦ 𝐷)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2146 ∃wrex 3091 Vcvv 3457 ⦋csb 3854 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4564 ↦ cmpt 5194 ^◡ccnv 5662 ran crn 5664 ↾ cres 5665 “ cima 5666 Fun wfun 6534 ⟶wf 6536 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 supcsup 9407 ℂcc 11113 ℝcr 11114 0cc0 11115 +∞cpnf 11255 ℝ^*cxr 11257 < clt 11258 [,)cico 13390 [,]cicc 13391 Σcsu 15761 Σ^{^}csumge0 47134

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-n0 12520 df-z 12607 df-uz 12879 df-rp 13033 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-clim 15563 df-sum 15762 df-sumge0 47135

This theorem is used by: sge0resrnlem 47175 sge0fodjrnlem 47188 sge0xp 47201 meadjiunlem 47237 isomenndlem 47302 ovnsubaddlem1 47342

W3C validator