Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0f1o Structured version   Visualization version   GIF version

Theorem sge0f1o 46337
Description: Re-index a nonnegative extended sum using a bijection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0f1o.1 𝑘𝜑
sge0f1o.2 𝑛𝜑
sge0f1o.3 (𝑘 = 𝐺𝐵 = 𝐷)
sge0f1o.4 (𝜑𝐶𝑉)
sge0f1o.5 (𝜑𝐹:𝐶1-1-onto𝐴)
sge0f1o.6 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
sge0f1o.7 ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
Assertion
Ref Expression
sge0f1o (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
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.
StepHypRef Expression
1 sge0f1o.4 . . . . . 6 (𝜑𝐶𝑉)
2 sge0f1o.5 . . . . . . 7 (𝜑𝐹:𝐶1-1-onto𝐴)
3 f1ofo 6855 . . . . . . 7 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶onto𝐴)
42, 3syl 17 . . . . . 6 (𝜑𝐹:𝐶onto𝐴)
5 focdmex 7978 . . . . . 6 (𝐶𝑉 → (𝐹:𝐶onto𝐴𝐴 ∈ V))
61, 4, 5sylc 65 . . . . 5 (𝜑𝐴 ∈ V)
76adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
8 sge0f1o.1 . . . . . 6 𝑘𝜑
9 sge0f1o.7 . . . . . 6 ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
108, 9fmptd2f 45177 . . . . 5 (𝜑 → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
1110adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
12 sge0f1o.2 . . . . . 6 𝑛𝜑
13 nfv 1911 . . . . . 6 𝑛+∞ ∈ ran (𝑘𝐴𝐵)
14 simp3 1137 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = 𝐷)
15 f1of 6848 . . . . . . . . . . . . . 14 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
162, 15syl 17 . . . . . . . . . . . . 13 (𝜑𝐹:𝐶𝐴)
1716ffvelcdmda 7103 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
18 sge0f1o.6 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
19 nfv 1911 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛) = 𝐺
20 nfcsb1v 3932 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) / 𝑘𝐵
2120nfeq1 2918 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛) / 𝑘𝐵 = 𝐷
2219, 21nfim 1893 . . . . . . . . . . . . 13 𝑘((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)
23 eqeq1 2738 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → (𝑘 = 𝐺 ↔ (𝐹𝑛) = 𝐺))
24 csbeq1a 3921 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
2524eqeq1d 2736 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → (𝐵 = 𝐷(𝐹𝑛) / 𝑘𝐵 = 𝐷))
2623, 25imbi12d 344 . . . . . . . . . . . . 13 (𝑘 = (𝐹𝑛) → ((𝑘 = 𝐺𝐵 = 𝐷) ↔ ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)))
27 sge0f1o.3 . . . . . . . . . . . . 13 (𝑘 = 𝐺𝐵 = 𝐷)
2822, 26, 27vtoclg1f 3569 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ 𝐴 → ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷))
2917, 18, 28sylc 65 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
3029eqcomd 2740 . . . . . . . . . 10 ((𝜑𝑛𝐶) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
31303adant3 1131 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
3214, 31eqtrd 2774 . . . . . . . 8 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = (𝐹𝑛) / 𝑘𝐵)
33 simpl 482 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → 𝜑)
3433, 17jca 511 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → (𝜑 ∧ (𝐹𝑛) ∈ 𝐴))
35 nfv 1911 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛) ∈ 𝐴
368, 35nfan 1896 . . . . . . . . . . . . 13 𝑘(𝜑 ∧ (𝐹𝑛) ∈ 𝐴)
3720nfel1 2919 . . . . . . . . . . . . 13 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)
3836, 37nfim 1893 . . . . . . . . . . . 12 𝑘((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
39 eleq1 2826 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → (𝑘𝐴 ↔ (𝐹𝑛) ∈ 𝐴))
4039anbi2d 630 . . . . . . . . . . . . 13 (𝑘 = (𝐹𝑛) → ((𝜑𝑘𝐴) ↔ (𝜑 ∧ (𝐹𝑛) ∈ 𝐴)))
4124eleq1d 2823 . . . . . . . . . . . . 13 (𝑘 = (𝐹𝑛) → (𝐵 ∈ (0[,]+∞) ↔ (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
4240, 41imbi12d 344 . . . . . . . . . . . 12 (𝑘 = (𝐹𝑛) → (((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))))
4338, 42, 9vtoclg1f 3569 . . . . . . . . . . 11 ((𝐹𝑛) ∈ 𝐴 → ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
4417, 34, 43sylc 65 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
45 eqid 2734 . . . . . . . . . . 11 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
4620, 45, 24elrnmpt1sf 45131 . . . . . . . . . 10 (((𝐹𝑛) ∈ 𝐴(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
4717, 44, 46syl2anc 584 . . . . . . . . 9 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
48473adant3 1131 . . . . . . . 8 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
4932, 48eqeltrd 2838 . . . . . . 7 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ ∈ ran (𝑘𝐴𝐵))
50493exp 1118 . . . . . 6 (𝜑 → (𝑛𝐶 → (+∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵))))
5112, 13, 50rexlimd 3263 . . . . 5 (𝜑 → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
52 pnfex 11311 . . . . . . 7 +∞ ∈ V
53 eqid 2734 . . . . . . . 8 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
5453elrnmpt 5971 . . . . . . 7 (+∞ ∈ V → (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷))
5552, 54ax-mp 5 . . . . . 6 (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷)
5655biimpi 216 . . . . 5 (+∞ ∈ ran (𝑛𝐶𝐷) → ∃𝑛𝐶 +∞ = 𝐷)
5751, 56impel 505 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑘𝐴𝐵))
587, 11, 57sge0pnfval 46328 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = +∞)
591adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
6030, 44eqeltrd 2838 . . . . . 6 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
6112, 60fmptd2f 45177 . . . . 5 (𝜑 → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
6261adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
63 simpr 484 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑛𝐶𝐷))
6459, 62, 63sge0pnfval 46328 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = +∞)
6558, 64eqtr4d 2777 . 2 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
66 sumex 15720 . . . . . . 7 Σ𝑘𝑦 𝐵 ∈ V
6766a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 ∈ V)
68 cnvimass 6101 . . . . . . . . . . . 12 (𝐹𝑦) ⊆ dom 𝐹
6968, 16fssdm 6755 . . . . . . . . . . 11 (𝜑 → (𝐹𝑦) ⊆ 𝐶)
701, 69sselpwd 5333 . . . . . . . . . 10 (𝜑 → (𝐹𝑦) ∈ 𝒫 𝐶)
7170adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ 𝒫 𝐶)
72 f1ocnv 6860 . . . . . . . . . . . 12 (𝐹:𝐶1-1-onto𝐴𝐹:𝐴1-1-onto𝐶)
732, 72syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝐴1-1-onto𝐶)
74 f1ofun 6850 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐶 → Fun 𝐹)
7573, 74syl 17 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
76 elinel2 4211 . . . . . . . . . 10 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
77 imafi 9350 . . . . . . . . . 10 ((Fun 𝐹𝑦 ∈ Fin) → (𝐹𝑦) ∈ Fin)
7875, 76, 77syl2an 596 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
7971, 78elind 4209 . . . . . . . 8 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
8079adantlr 715 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
81 nfv 1911 . . . . . . . . . 10 𝑘 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
828, 81nfan 1896 . . . . . . . . 9 𝑘(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
83 nfv 1911 . . . . . . . . 9 𝑘 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
8482, 83nfan 1896 . . . . . . . 8 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
85 nfmpt1 5255 . . . . . . . . . . . . 13 𝑛(𝑛𝐶𝐷)
8685nfrn 5965 . . . . . . . . . . . 12 𝑛ran (𝑛𝐶𝐷)
8786nfel2 2921 . . . . . . . . . . 11 𝑛+∞ ∈ ran (𝑛𝐶𝐷)
8887nfn 1854 . . . . . . . . . 10 𝑛 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
8912, 88nfan 1896 . . . . . . . . 9 𝑛(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
90 nfv 1911 . . . . . . . . 9 𝑛 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
9189, 90nfan 1896 . . . . . . . 8 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
9278adantlr 715 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
93 f1of1 6847 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1𝐴)
942, 93syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐶1-1𝐴)
9594adantr 480 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
9669adantr 480 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ⊆ 𝐶)
97 f1ores 6862 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴 ∧ (𝐹𝑦) ⊆ 𝐶) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
9895, 96, 97syl2anc 584 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
99 elpwinss 44988 . . . . . . . . . . . 12 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦𝐴)
100 foimacnv 6865 . . . . . . . . . . . 12 ((𝐹:𝐶onto𝐴𝑦𝐴) → (𝐹 “ (𝐹𝑦)) = 𝑦)
1014, 99, 100syl2an 596 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
102101f1oeq3d 6845 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)) ↔ (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦))
10398, 102mpbid 232 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
104103adantlr 715 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
10516, 1fexd 7246 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
106 cnvexg 7946 . . . . . . . . . . . . 13 (𝐹 ∈ V → 𝐹 ∈ V)
107105, 106syl 17 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
108107imaexd 7938 . . . . . . . . . . 11 (𝜑 → (𝐹𝑦) ∈ V)
109108ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ V)
110 simpll 767 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝜑)
11179adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
112 simpr 484 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐹𝑦))
113110, 111, 112jca31 514 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)))
114 eleq1 2826 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)))
115114anbi2d 630 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ (𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
116 eleq2w2 2730 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝑛𝑥𝑛 ∈ (𝐹𝑦)))
117115, 116anbi12d 632 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) ↔ ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦))))
118 reseq2 5994 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 ↾ (𝐹𝑦)))
119118fveq1d 6908 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝐹𝑥)‘𝑛) = ((𝐹 ↾ (𝐹𝑦))‘𝑛))
120119eqeq1d 2736 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝐹𝑥)‘𝑛) = 𝐺 ↔ ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
121117, 120imbi12d 344 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺) ↔ (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)))
122 fvres 6925 . . . . . . . . . . . . 13 (𝑛𝑥 → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
123122adantl 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
124 simpll 767 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝜑)
125 elpwinss 44988 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥𝐶)
126125adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥𝐶)
127126sselda 3994 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝑛𝐶)
128124, 127, 18syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → (𝐹𝑛) = 𝐺)
129123, 128eqtrd 2774 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
130121, 129vtoclg 3553 . . . . . . . . . 10 ((𝐹𝑦) ∈ V → (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
131109, 113, 130sylc 65 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
132131adantllr 719 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
133108ad3antrrr 730 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ V)
134 simpll 767 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)))
13570ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ 𝒫 𝐶)
13692adantr 480 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ Fin)
137135, 136elind 4209 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
138 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘𝑦)
139101eqcomd 2740 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 = (𝐹 “ (𝐹𝑦)))
140139adantr 480 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑦 = (𝐹 “ (𝐹𝑦)))
141138, 140eleqtrd 2840 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
142141adantllr 719 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
143134, 137, 142jca31 514 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
144114anbi2d 630 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
145 imaeq2 6075 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
146145eleq2d 2824 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (𝑘 ∈ (𝐹𝑥) ↔ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
147144, 146anbi12d 632 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) ↔ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦)))))
148147imbi1d 341 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → (((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ) ↔ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))) → 𝐵 ∈ ℂ)))
149 rge0ssre 13492 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
150 ax-resscn 11209 . . . . . . . . . . . 12 ℝ ⊆ ℂ
151149, 150sstri 4004 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℂ
152 simplll 775 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝜑)
153 simpllr 776 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
15416fimassd 6757 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑥) ⊆ 𝐴)
155154ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → (𝐹𝑥) ⊆ 𝐴)
156 simpr 484 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘 ∈ (𝐹𝑥))
157155, 156sseldd 3995 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
158157adantllr 719 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
159 foelcdmi 6969 . . . . . . . . . . . . . . 15 ((𝐹:𝐶onto𝐴𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
1604, 159sylan 580 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
161160adantlr 715 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
162 nfv 1911 . . . . . . . . . . . . . . 15 𝑛 𝑘𝐴
16389, 162nfan 1896 . . . . . . . . . . . . . 14 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴)
164 nfv 1911 . . . . . . . . . . . . . 14 𝑛 𝐵 ∈ (0[,)+∞)
165 csbid 3920 . . . . . . . . . . . . . . . . . . . . 21 𝑘 / 𝑘𝐵 = 𝐵
166165eqcomi 2743 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 𝑘 / 𝑘𝐵
167166a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝑘 / 𝑘𝐵)
168 id 22 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑛) = 𝑘 → (𝐹𝑛) = 𝑘)
169168eqcomd 2740 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑛) = 𝑘𝑘 = (𝐹𝑛))
170169csbeq1d 3911 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑛) = 𝑘𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
1711703ad2ant3 1134 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
172293adant3 1131 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
173167, 171, 1723eqtrd 2778 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
1741733adant1r 1176 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
175 0xr 11305 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℝ*
176175a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 0 ∈ ℝ*)
177 pnfxr 11312 . . . . . . . . . . . . . . . . . . . . . . . 24 +∞ ∈ ℝ*
178177a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ℝ*)
17960adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ (0[,]+∞))
180 simpr 484 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ 𝐷 ∈ (0[,)+∞))
181176, 178, 179, 180eliccnelico 45481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 = +∞)
182181eqcomd 2740 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ = 𝐷)
183 simpr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛𝐶) → 𝑛𝐶)
18453, 183, 60elrnmpt1d 5977 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛𝐶) → 𝐷 ∈ ran (𝑛𝐶𝐷))
185184adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
186182, 185eqeltrd 2838 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
187186adantllr 719 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
188 simpllr 776 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
189187, 188condan 818 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) → 𝐷 ∈ (0[,)+∞))
1901893adant3 1131 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐷 ∈ (0[,)+∞))
191174, 190eqeltrd 2838 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 ∈ (0[,)+∞))
1921913exp 1118 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
193192adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
194163, 164, 193rexlimd 3263 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (∃𝑛𝐶 (𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞)))
195161, 194mpd 15 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → 𝐵 ∈ (0[,)+∞))
196152, 153, 158, 195syl21anc 838 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ (0[,)+∞))
197151, 196sselid 3992 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
198148, 197vtoclg 3553 . . . . . . . . 9 ((𝐹𝑦) ∈ V → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))) → 𝐵 ∈ ℂ))
199133, 143, 198sylc 65 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
20084, 91, 27, 92, 104, 132, 199fsumf1of 45529 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
201 sumeq1 15721 . . . . . . . 8 (𝑥 = (𝐹𝑦) → Σ𝑛𝑥 𝐷 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
202201rspceeqv 3644 . . . . . . 7 (((𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
20380, 200, 202syl2anc 584 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
20467, 203rnmptssrn 45124 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
205 sumex 15720 . . . . . . 7 Σ𝑛𝑥 𝐷 ∈ V
206205a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 ∈ V)
2076, 154sselpwd 5333 . . . . . . . . . 10 (𝜑 → (𝐹𝑥) ∈ 𝒫 𝐴)
208207adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ 𝒫 𝐴)
20916ffund 6740 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
210 elinel2 4211 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ∈ Fin)
211 imafi 9350 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ Fin) → (𝐹𝑥) ∈ Fin)
212209, 210, 211syl2an 596 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ Fin)
213208, 212elind 4209 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
214213adantlr 715 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
215 nfv 1911 . . . . . . . . . 10 𝑘 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
21682, 215nfan 1896 . . . . . . . . 9 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
217 nfv 1911 . . . . . . . . . 10 𝑛 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
21889, 217nfan 1896 . . . . . . . . 9 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
219210adantl 481 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
220 f1ores 6862 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴𝑥𝐶) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
22194, 125, 220syl2an 596 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
222221adantlr 715 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
223129adantllr 719 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
224216, 218, 27, 219, 222, 223, 197fsumf1of 45529 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑘 ∈ (𝐹𝑥)𝐵 = Σ𝑛𝑥 𝐷)
225224eqcomd 2740 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
226 sumeq1 15721 . . . . . . . 8 (𝑦 = (𝐹𝑥) → Σ𝑘𝑦 𝐵 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
227226rspceeqv 3644 . . . . . . 7 (((𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
228214, 225, 227syl2anc 584 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
229206, 228rnmptssrn 45124 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷) ⊆ ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵))
230204, 229eqssd 4012 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) = ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
231230supeq1d 9483 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
2326adantr 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
23382, 232, 195sge0revalmpt 46333 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ))
2341adantr 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
23589, 234, 189sge0revalmpt 46333 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
236231, 233, 2353eqtr4d 2784 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
23765, 236pm2.61dan 813 1 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wnf 1779  wcel 2105  wrex 3067  Vcvv 3477  csb 3907  cin 3961  wss 3962  𝒫 cpw 4604  cmpt 5230  ccnv 5687  ran crn 5689  cres 5690  cima 5691  Fun wfun 6556  wf 6558  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  Fincfn 8983  supcsup 9477  cc 11150  cr 11151  0cc0 11152  +∞cpnf 11289  *cxr 11291   < clt 11292  [,)cico 13385  [,]cicc 13386  Σcsu 15718  Σ^csumge0 46317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-sumge0 46318
This theorem is referenced by:  sge0resrnlem  46358  sge0fodjrnlem  46371  sge0xp  46384  meadjiunlem  46420  isomenndlem  46485  ovnsubaddlem1  46525
  Copyright terms: Public domain W3C validator