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Theorem sge0f1o 42206
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 6490 . . . . . . 7 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶onto𝐴)
42, 3syl 17 . . . . . 6 (𝜑𝐹:𝐶onto𝐴)
5 fornex 7513 . . . . . 6 (𝐶𝑉 → (𝐹:𝐶onto𝐴𝐴 ∈ V))
61, 4, 5sylc 65 . . . . 5 (𝜑𝐴 ∈ V)
76adantr 481 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
8 sge0f1o.1 . . . . . 6 𝑘𝜑
9 sge0f1o.7 . . . . . 6 ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
10 eqid 2795 . . . . . 6 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
118, 9, 10fmptdf 6744 . . . . 5 (𝜑 → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
1211adantr 481 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
13 pnfex 10540 . . . . . . . 8 +∞ ∈ V
14 eqid 2795 . . . . . . . . 9 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
1514elrnmpt 5710 . . . . . . . 8 (+∞ ∈ V → (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷))
1613, 15ax-mp 5 . . . . . . 7 (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷)
1716biimpi 217 . . . . . 6 (+∞ ∈ ran (𝑛𝐶𝐷) → ∃𝑛𝐶 +∞ = 𝐷)
1817adantl 482 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → ∃𝑛𝐶 +∞ = 𝐷)
19 sge0f1o.2 . . . . . . 7 𝑛𝜑
20 nfv 1892 . . . . . . 7 𝑛+∞ ∈ ran (𝑘𝐴𝐵)
21 simp3 1131 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = 𝐷)
22 f1of 6483 . . . . . . . . . . . . . . 15 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
232, 22syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶𝐴)
2423ffvelrnda 6716 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
25 sge0f1o.6 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
26 nfcv 2949 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛)
27 nfv 1892 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) = 𝐺
2826nfcsb1 3832 . . . . . . . . . . . . . . . 16 𝑘(𝐹𝑛) / 𝑘𝐵
29 nfcv 2949 . . . . . . . . . . . . . . . 16 𝑘𝐷
3028, 29nfeq 2960 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) / 𝑘𝐵 = 𝐷
3127, 30nfim 1878 . . . . . . . . . . . . . 14 𝑘((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)
32 eqeq1 2799 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘 = 𝐺 ↔ (𝐹𝑛) = 𝐺))
33 csbeq1a 3824 . . . . . . . . . . . . . . . 16 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
3433eqeq1d 2797 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝐵 = 𝐷(𝐹𝑛) / 𝑘𝐵 = 𝐷))
3532, 34imbi12d 346 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → ((𝑘 = 𝐺𝐵 = 𝐷) ↔ ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)))
36 sge0f1o.3 . . . . . . . . . . . . . 14 (𝑘 = 𝐺𝐵 = 𝐷)
3726, 31, 35, 36vtoclgf 3508 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ 𝐴 → ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷))
3824, 25, 37sylc 65 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
3938eqcomd 2801 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
40393adant3 1125 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
4121, 40eqtrd 2831 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = (𝐹𝑛) / 𝑘𝐵)
42 simpl 483 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → 𝜑)
4342, 24jca 512 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝜑 ∧ (𝐹𝑛) ∈ 𝐴))
44 nfv 1892 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) ∈ 𝐴
458, 44nfan 1881 . . . . . . . . . . . . . 14 𝑘(𝜑 ∧ (𝐹𝑛) ∈ 𝐴)
4628nfel1 2963 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)
4745, 46nfim 1878 . . . . . . . . . . . . 13 𝑘((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
48 eleq1 2870 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘𝐴 ↔ (𝐹𝑛) ∈ 𝐴))
4948anbi2d 628 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → ((𝜑𝑘𝐴) ↔ (𝜑 ∧ (𝐹𝑛) ∈ 𝐴)))
5033eleq1d 2867 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → (𝐵 ∈ (0[,]+∞) ↔ (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
5149, 50imbi12d 346 . . . . . . . . . . . . 13 (𝑘 = (𝐹𝑛) → (((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))))
5226, 47, 51, 9vtoclgf 3508 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ 𝐴 → ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
5324, 43, 52sylc 65 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
5428, 10, 33elrnmpt1sf 40990 . . . . . . . . . . 11 (((𝐹𝑛) ∈ 𝐴(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5524, 53, 54syl2anc 584 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
56553adant3 1125 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5741, 56eqeltrd 2883 . . . . . . . 8 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ ∈ ran (𝑘𝐴𝐵))
58573exp 1112 . . . . . . 7 (𝜑 → (𝑛𝐶 → (+∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵))))
5919, 20, 58rexlimd 3278 . . . . . 6 (𝜑 → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6059adantr 481 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6118, 60mpd 15 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑘𝐴𝐵))
627, 12, 61sge0pnfval 42197 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = +∞)
631adantr 481 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
6439, 53eqeltrd 2883 . . . . . 6 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
6519, 64, 14fmptdf 6744 . . . . 5 (𝜑 → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
6665adantr 481 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
67 simpr 485 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑛𝐶𝐷))
6863, 66, 67sge0pnfval 42197 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = +∞)
6962, 68eqtr4d 2834 . 2 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
70 sumex 14878 . . . . . . 7 Σ𝑘𝑦 𝐵 ∈ V
7170a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 ∈ V)
72 cnvimass 5825 . . . . . . . . . . . 12 (𝐹𝑦) ⊆ dom 𝐹
7372, 23fssdm 6398 . . . . . . . . . . 11 (𝜑 → (𝐹𝑦) ⊆ 𝐶)
74 fex 6855 . . . . . . . . . . . . . . 15 ((𝐹:𝐶𝐴𝐶𝑉) → 𝐹 ∈ V)
7523, 1, 74syl2anc 584 . . . . . . . . . . . . . 14 (𝜑𝐹 ∈ V)
76 cnvexg 7485 . . . . . . . . . . . . . 14 (𝐹 ∈ V → 𝐹 ∈ V)
7775, 76syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
78 imaexg 7476 . . . . . . . . . . . . 13 (𝐹 ∈ V → (𝐹𝑦) ∈ V)
7977, 78syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑦) ∈ V)
80 elpwg 4461 . . . . . . . . . . . 12 ((𝐹𝑦) ∈ V → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8179, 80syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8273, 81mpbird 258 . . . . . . . . . 10 (𝜑 → (𝐹𝑦) ∈ 𝒫 𝐶)
8382adantr 481 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ 𝒫 𝐶)
84 f1ocnv 6495 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐴1-1-onto𝐶)
852, 84syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐴1-1-onto𝐶)
86 f1ofun 6485 . . . . . . . . . . . 12 (𝐹:𝐴1-1-onto𝐶 → Fun 𝐹)
8785, 86syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
8887adantr 481 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Fun 𝐹)
89 elinel2 4094 . . . . . . . . . . 11 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
9089adantl 482 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
91 imafi 8663 . . . . . . . . . 10 ((Fun 𝐹𝑦 ∈ Fin) → (𝐹𝑦) ∈ Fin)
9288, 90, 91syl2anc 584 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
9383, 92elind 4092 . . . . . . . 8 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
9493adantlr 711 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
95 nfv 1892 . . . . . . . . . 10 𝑘 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
968, 95nfan 1881 . . . . . . . . 9 𝑘(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
97 nfv 1892 . . . . . . . . 9 𝑘 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
9896, 97nfan 1881 . . . . . . . 8 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
99 nfcv 2949 . . . . . . . . . . . 12 𝑛+∞
100 nfmpt1 5058 . . . . . . . . . . . . 13 𝑛(𝑛𝐶𝐷)
101100nfrn 5706 . . . . . . . . . . . 12 𝑛ran (𝑛𝐶𝐷)
10299, 101nfel 2961 . . . . . . . . . . 11 𝑛+∞ ∈ ran (𝑛𝐶𝐷)
103102nfn 1838 . . . . . . . . . 10 𝑛 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
10419, 103nfan 1881 . . . . . . . . 9 𝑛(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
105 nfv 1892 . . . . . . . . 9 𝑛 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
106104, 105nfan 1881 . . . . . . . 8 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
10792adantlr 711 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
108 f1of1 6482 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1𝐴)
1092, 108syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐶1-1𝐴)
110109adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
11181adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
11283, 111mpbid 233 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ⊆ 𝐶)
113 f1ores 6497 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴 ∧ (𝐹𝑦) ⊆ 𝐶) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
114110, 112, 113syl2anc 584 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
1154adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶onto𝐴)
116 elpwinss 40850 . . . . . . . . . . . . 13 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦𝐴)
117116adantl 482 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦𝐴)
118 foimacnv 6500 . . . . . . . . . . . 12 ((𝐹:𝐶onto𝐴𝑦𝐴) → (𝐹 “ (𝐹𝑦)) = 𝑦)
119115, 117, 118syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
120119f1oeq3d 6480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)) ↔ (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦))
121114, 120mpbid 233 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
122121adantlr 711 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
12379ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ V)
124 simpll 763 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝜑)
12593adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
126 simpr 485 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐹𝑦))
127124, 125, 126jca31 515 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)))
128 eleq1 2870 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)))
129128anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ (𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
130 eleq2 2871 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝑛𝑥𝑛 ∈ (𝐹𝑦)))
131129, 130anbi12d 630 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) ↔ ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦))))
132 reseq2 5729 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 ↾ (𝐹𝑦)))
133132fveq1d 6540 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝐹𝑥)‘𝑛) = ((𝐹 ↾ (𝐹𝑦))‘𝑛))
134133eqeq1d 2797 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝐹𝑥)‘𝑛) = 𝐺 ↔ ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
135131, 134imbi12d 346 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺) ↔ (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)))
136 fvres 6557 . . . . . . . . . . . . 13 (𝑛𝑥 → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
137136adantl 482 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
138 simpll 763 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝜑)
139 elpwinss 40850 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥𝐶)
140139adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥𝐶)
141140sselda 3889 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝑛𝐶)
142138, 141, 25syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → (𝐹𝑛) = 𝐺)
143137, 142eqtrd 2831 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
144135, 143vtoclg 3510 . . . . . . . . . 10 ((𝐹𝑦) ∈ V → (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
145123, 127, 144sylc 65 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
146145adantllr 715 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
14779ad3antrrr 726 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ V)
148 simpll 763 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)))
14982ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ 𝒫 𝐶)
150107adantr 481 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ Fin)
151149, 150elind 4092 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
152 simpr 485 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘𝑦)
153119eqcomd 2801 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 = (𝐹 “ (𝐹𝑦)))
154153adantr 481 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑦 = (𝐹 “ (𝐹𝑦)))
155152, 154eleqtrd 2885 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
156155adantllr 715 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
157148, 151, 156jca31 515 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
158128anbi2d 628 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
159 imaeq2 5802 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
160159eleq2d 2868 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (𝑘 ∈ (𝐹𝑥) ↔ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
161158, 160anbi12d 630 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) ↔ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦)))))
162161imbi1d 343 . . . . . . . . . 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)))
163 rge0ssre 12694 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
164 ax-resscn 10440 . . . . . . . . . . . . 13 ℝ ⊆ ℂ
165163, 164sstri 3898 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℂ
166 simplll 771 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝜑)
167 simpllr 772 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
168 fimass 6423 . . . . . . . . . . . . . . . . 17 (𝐹:𝐶𝐴 → (𝐹𝑥) ⊆ 𝐴)
16923, 168syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹𝑥) ⊆ 𝐴)
170169ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → (𝐹𝑥) ⊆ 𝐴)
171 simpr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘 ∈ (𝐹𝑥))
172170, 171sseldd 3890 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
173172adantllr 715 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
174 foelrni 6595 . . . . . . . . . . . . . . . 16 ((𝐹:𝐶onto𝐴𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
1754, 174sylan 580 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
176175adantlr 711 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
177 nfv 1892 . . . . . . . . . . . . . . . 16 𝑛 𝑘𝐴
178104, 177nfan 1881 . . . . . . . . . . . . . . 15 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴)
179 nfv 1892 . . . . . . . . . . . . . . 15 𝑛 𝐵 ∈ (0[,)+∞)
180 csbid 3823 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 / 𝑘𝐵 = 𝐵
181180eqcomi 2804 . . . . . . . . . . . . . . . . . . . . 21 𝐵 = 𝑘 / 𝑘𝐵
182181a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝑘 / 𝑘𝐵)
183 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) = 𝑘 → (𝐹𝑛) = 𝑘)
184183eqcomd 2801 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑛) = 𝑘𝑘 = (𝐹𝑛))
185184csbeq1d 3815 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑛) = 𝑘𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
1861853ad2ant3 1128 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
18738idi 1 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
1881873adant3 1125 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
189182, 186, 1883eqtrd 2835 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
1901893adant1r 1170 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
191 0xr 10534 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ ℝ*
192191a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 0 ∈ ℝ*)
193 pnfxr 10541 . . . . . . . . . . . . . . . . . . . . . . . . 25 +∞ ∈ ℝ*
194193a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ℝ*)
19564adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ (0[,]+∞))
196 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ 𝐷 ∈ (0[,)+∞))
197192, 194, 195, 196eliccnelico 41347 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 = +∞)
198197eqcomd 2801 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ = 𝐷)
199 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝑛𝐶)
20064idi 1 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
20114elrnmpt1 5712 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝐶𝐷 ∈ (0[,]+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
202199, 200, 201syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛𝐶) → 𝐷 ∈ ran (𝑛𝐶𝐷))
203202adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
204198, 203eqeltrd 2883 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
205204adantllr 715 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
206 simpllr 772 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
207205, 206condan 814 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) → 𝐷 ∈ (0[,)+∞))
2082073adant3 1125 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐷 ∈ (0[,)+∞))
209190, 208eqeltrd 2883 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 ∈ (0[,)+∞))
2102093exp 1112 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
211210adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
212178, 179, 211rexlimd 3278 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (∃𝑛𝐶 (𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞)))
213176, 212mpd 15 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → 𝐵 ∈ (0[,)+∞))
214166, 167, 173, 213syl21anc 834 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ (0[,)+∞))
215165, 214sseldi 3887 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
216215idi 1 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
217162, 216vtoclg 3510 . . . . . . . . 9 ((𝐹𝑦) ∈ V → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))) → 𝐵 ∈ ℂ))
218147, 157, 217sylc 65 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
21998, 106, 36, 107, 122, 146, 218fsumf1of 41397 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
220 sumeq1 14879 . . . . . . . 8 (𝑥 = (𝐹𝑦) → Σ𝑛𝑥 𝐷 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
221220rspceeqv 3577 . . . . . . 7 (((𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22294, 219, 221syl2anc 584 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22371, 222rnmptssrn 40982 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
224 sumex 14878 . . . . . . 7 Σ𝑛𝑥 𝐷 ∈ V
225224a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 ∈ V)
2266, 169ssexd 5119 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑥) ∈ V)
227 elpwg 4461 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
228226, 227syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
229169, 228mpbird 258 . . . . . . . . . 10 (𝜑 → (𝐹𝑥) ∈ 𝒫 𝐴)
230229adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ 𝒫 𝐴)
23123ffund 6386 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
232231adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Fun 𝐹)
233 elinel2 4094 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ∈ Fin)
234233adantl 482 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
235 imafi 8663 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ Fin) → (𝐹𝑥) ∈ Fin)
236232, 234, 235syl2anc 584 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ Fin)
237230, 236elind 4092 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
238237adantlr 711 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
239 nfv 1892 . . . . . . . . . 10 𝑘 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
24096, 239nfan 1881 . . . . . . . . 9 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
241 nfv 1892 . . . . . . . . . 10 𝑛 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
242104, 241nfan 1881 . . . . . . . . 9 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
243233adantl 482 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
244109adantr 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
245 f1ores 6497 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴𝑥𝐶) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
246244, 140, 245syl2anc 584 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
247246adantlr 711 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
248143adantllr 715 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
249240, 242, 36, 243, 247, 248, 215fsumf1of 41397 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑘 ∈ (𝐹𝑥)𝐵 = Σ𝑛𝑥 𝐷)
250249eqcomd 2801 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
251 sumeq1 14879 . . . . . . . 8 (𝑦 = (𝐹𝑥) → Σ𝑘𝑦 𝐵 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
252251rspceeqv 3577 . . . . . . 7 (((𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
253238, 250, 252syl2anc 584 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
254225, 253rnmptssrn 40982 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷) ⊆ ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵))
255223, 254eqssd 3906 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) = ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
256255supeq1d 8756 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
2576adantr 481 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
25896, 257, 213sge0revalmpt 42202 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ))
2591adantr 481 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
260104, 259, 207sge0revalmpt 42202 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
261256, 258, 2603eqtr4d 2841 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
26269, 261pm2.61dan 809 1 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wnf 1765  wcel 2081  wrex 3106  Vcvv 3437  csb 3811  cin 3858  wss 3859  𝒫 cpw 4453  cmpt 5041  ccnv 5442  ran crn 5444  cres 5445  cima 5446  Fun wfun 6219  wf 6221  1-1wf1 6222  ontowfo 6223  1-1-ontowf1o 6224  cfv 6225  (class class class)co 7016  Fincfn 8357  supcsup 8750  cc 10381  cr 10382  0cc0 10383  +∞cpnf 10518  *cxr 10520   < clt 10521  [,)cico 12590  [,]cicc 12591  Σcsu 14876  Σ^csumge0 42186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-sum 14877  df-sumge0 42187
This theorem is referenced by:  sge0resrnlem  42227  sge0fodjrnlem  42240  sge0xp  42253  meadjiunlem  42289  isomenndlem  42354  ovnsubaddlem1  42394
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