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Theorem sge0f1o 41113
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 6286 . . . . . . 7 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶onto𝐴)
42, 3syl 17 . . . . . 6 (𝜑𝐹:𝐶onto𝐴)
5 fornex 7285 . . . . . 6 (𝐶𝑉 → (𝐹:𝐶onto𝐴𝐴 ∈ V))
61, 4, 5sylc 65 . . . . 5 (𝜑𝐴 ∈ V)
76adantr 466 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
8 sge0f1o.1 . . . . . 6 𝑘𝜑
9 sge0f1o.7 . . . . . 6 ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
10 eqid 2771 . . . . . 6 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
118, 9, 10fmptdf 6531 . . . . 5 (𝜑 → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
1211adantr 466 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
13 pnfex 10298 . . . . . . . 8 +∞ ∈ V
14 eqid 2771 . . . . . . . . 9 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
1514elrnmpt 5509 . . . . . . . 8 (+∞ ∈ V → (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷))
1613, 15ax-mp 5 . . . . . . 7 (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷)
1716biimpi 206 . . . . . 6 (+∞ ∈ ran (𝑛𝐶𝐷) → ∃𝑛𝐶 +∞ = 𝐷)
1817adantl 467 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → ∃𝑛𝐶 +∞ = 𝐷)
19 sge0f1o.2 . . . . . . 7 𝑛𝜑
20 nfv 1995 . . . . . . 7 𝑛+∞ ∈ ran (𝑘𝐴𝐵)
21 simp3 1132 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = 𝐷)
22 f1of 6279 . . . . . . . . . . . . . . 15 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
232, 22syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶𝐴)
2423ffvelrnda 6504 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
25 sge0f1o.6 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
26 nfcv 2913 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛)
27 nfv 1995 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) = 𝐺
2826nfcsb1 3697 . . . . . . . . . . . . . . . 16 𝑘(𝐹𝑛) / 𝑘𝐵
29 nfcv 2913 . . . . . . . . . . . . . . . 16 𝑘𝐷
3028, 29nfeq 2925 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) / 𝑘𝐵 = 𝐷
3127, 30nfim 1977 . . . . . . . . . . . . . 14 𝑘((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)
32 eqeq1 2775 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘 = 𝐺 ↔ (𝐹𝑛) = 𝐺))
33 csbeq1a 3691 . . . . . . . . . . . . . . . 16 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
3433eqeq1d 2773 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝐵 = 𝐷(𝐹𝑛) / 𝑘𝐵 = 𝐷))
3532, 34imbi12d 333 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → ((𝑘 = 𝐺𝐵 = 𝐷) ↔ ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)))
36 sge0f1o.3 . . . . . . . . . . . . . 14 (𝑘 = 𝐺𝐵 = 𝐷)
3726, 31, 35, 36vtoclgf 3415 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ 𝐴 → ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷))
3824, 25, 37sylc 65 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
3938eqcomd 2777 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
40393adant3 1126 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
4121, 40eqtrd 2805 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = (𝐹𝑛) / 𝑘𝐵)
42 simpl 468 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → 𝜑)
4342, 24jca 501 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝜑 ∧ (𝐹𝑛) ∈ 𝐴))
44 nfv 1995 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) ∈ 𝐴
458, 44nfan 1980 . . . . . . . . . . . . . 14 𝑘(𝜑 ∧ (𝐹𝑛) ∈ 𝐴)
4628nfel1 2928 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)
4745, 46nfim 1977 . . . . . . . . . . . . 13 𝑘((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
48 eleq1 2838 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘𝐴 ↔ (𝐹𝑛) ∈ 𝐴))
4948anbi2d 614 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → ((𝜑𝑘𝐴) ↔ (𝜑 ∧ (𝐹𝑛) ∈ 𝐴)))
5033eleq1d 2835 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → (𝐵 ∈ (0[,]+∞) ↔ (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
5149, 50imbi12d 333 . . . . . . . . . . . . 13 (𝑘 = (𝐹𝑛) → (((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))))
5226, 47, 51, 9vtoclgf 3415 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ 𝐴 → ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
5324, 43, 52sylc 65 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
5428, 10, 33elrnmpt1sf 39895 . . . . . . . . . . 11 (((𝐹𝑛) ∈ 𝐴(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5524, 53, 54syl2anc 573 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
56553adant3 1126 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5741, 56eqeltrd 2850 . . . . . . . 8 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ ∈ ran (𝑘𝐴𝐵))
58573exp 1112 . . . . . . 7 (𝜑 → (𝑛𝐶 → (+∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵))))
5919, 20, 58rexlimd 3174 . . . . . 6 (𝜑 → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6059adantr 466 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6118, 60mpd 15 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑘𝐴𝐵))
627, 12, 61sge0pnfval 41104 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = +∞)
631adantr 466 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
6439, 53eqeltrd 2850 . . . . . 6 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
6519, 64, 14fmptdf 6531 . . . . 5 (𝜑 → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
6665adantr 466 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
67 simpr 471 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑛𝐶𝐷))
6863, 66, 67sge0pnfval 41104 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = +∞)
6962, 68eqtr4d 2808 . 2 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
70 sumex 14625 . . . . . . 7 Σ𝑘𝑦 𝐵 ∈ V
7170a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 ∈ V)
72 cnvimass 5625 . . . . . . . . . . . . 13 (𝐹𝑦) ⊆ dom 𝐹
7372a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑦) ⊆ dom 𝐹)
7423fdmd 6193 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐶)
7573, 74sseqtrd 3790 . . . . . . . . . . 11 (𝜑 → (𝐹𝑦) ⊆ 𝐶)
76 fex 6635 . . . . . . . . . . . . . . 15 ((𝐹:𝐶𝐴𝐶𝑉) → 𝐹 ∈ V)
7723, 1, 76syl2anc 573 . . . . . . . . . . . . . 14 (𝜑𝐹 ∈ V)
78 cnvexg 7262 . . . . . . . . . . . . . 14 (𝐹 ∈ V → 𝐹 ∈ V)
7977, 78syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
80 imaexg 7253 . . . . . . . . . . . . 13 (𝐹 ∈ V → (𝐹𝑦) ∈ V)
8179, 80syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑦) ∈ V)
82 elpwg 4306 . . . . . . . . . . . 12 ((𝐹𝑦) ∈ V → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8381, 82syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8475, 83mpbird 247 . . . . . . . . . 10 (𝜑 → (𝐹𝑦) ∈ 𝒫 𝐶)
8584adantr 466 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ 𝒫 𝐶)
86 f1ocnv 6291 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐴1-1-onto𝐶)
872, 86syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐴1-1-onto𝐶)
88 f1ofun 6281 . . . . . . . . . . . 12 (𝐹:𝐴1-1-onto𝐶 → Fun 𝐹)
8987, 88syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
9089adantr 466 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Fun 𝐹)
91 elinel2 3951 . . . . . . . . . . 11 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
9291adantl 467 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
93 imafi 8418 . . . . . . . . . 10 ((Fun 𝐹𝑦 ∈ Fin) → (𝐹𝑦) ∈ Fin)
9490, 92, 93syl2anc 573 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
9585, 94elind 3949 . . . . . . . 8 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
9695adantlr 694 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
97 nfv 1995 . . . . . . . . . 10 𝑘 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
988, 97nfan 1980 . . . . . . . . 9 𝑘(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
99 nfv 1995 . . . . . . . . 9 𝑘 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
10098, 99nfan 1980 . . . . . . . 8 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
101 nfcv 2913 . . . . . . . . . . . 12 𝑛+∞
102 nfmpt1 4882 . . . . . . . . . . . . 13 𝑛(𝑛𝐶𝐷)
103102nfrn 5505 . . . . . . . . . . . 12 𝑛ran (𝑛𝐶𝐷)
104101, 103nfel 2926 . . . . . . . . . . 11 𝑛+∞ ∈ ran (𝑛𝐶𝐷)
105104nfn 1935 . . . . . . . . . 10 𝑛 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
10619, 105nfan 1980 . . . . . . . . 9 𝑛(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
107 nfv 1995 . . . . . . . . 9 𝑛 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
108106, 107nfan 1980 . . . . . . . 8 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
10994adantlr 694 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
110 f1of1 6278 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1𝐴)
1112, 110syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐶1-1𝐴)
112111adantr 466 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
11383adantr 466 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
11485, 113mpbid 222 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ⊆ 𝐶)
115 f1ores 6293 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴 ∧ (𝐹𝑦) ⊆ 𝐶) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
116112, 114, 115syl2anc 573 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
1174adantr 466 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶onto𝐴)
118 elpwinss 39737 . . . . . . . . . . . . 13 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦𝐴)
119118adantl 467 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦𝐴)
120 foimacnv 6296 . . . . . . . . . . . 12 ((𝐹:𝐶onto𝐴𝑦𝐴) → (𝐹 “ (𝐹𝑦)) = 𝑦)
121117, 119, 120syl2anc 573 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
122121f1oeq3d 6276 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)) ↔ (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦))
123116, 122mpbid 222 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
124123adantlr 694 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
12581ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ V)
126 simpll 750 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝜑)
12795adantr 466 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
128 simpr 471 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐹𝑦))
129126, 127, 128jca31 504 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)))
130 eleq1 2838 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)))
131130anbi2d 614 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ (𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
132 eleq2 2839 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝑛𝑥𝑛 ∈ (𝐹𝑦)))
133131, 132anbi12d 616 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) ↔ ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦))))
134 reseq2 5528 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 ↾ (𝐹𝑦)))
135134fveq1d 6335 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝐹𝑥)‘𝑛) = ((𝐹 ↾ (𝐹𝑦))‘𝑛))
136135eqeq1d 2773 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝐹𝑥)‘𝑛) = 𝐺 ↔ ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
137133, 136imbi12d 333 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺) ↔ (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)))
138 fvres 6350 . . . . . . . . . . . . 13 (𝑛𝑥 → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
139138adantl 467 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
140 simpll 750 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝜑)
141 elpwinss 39737 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥𝐶)
142141adantl 467 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥𝐶)
143142sselda 3752 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝑛𝐶)
144140, 143, 25syl2anc 573 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → (𝐹𝑛) = 𝐺)
145139, 144eqtrd 2805 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
146137, 145vtoclg 3417 . . . . . . . . . 10 ((𝐹𝑦) ∈ V → (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
147125, 129, 146sylc 65 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
148147adantllr 698 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
14981ad3antrrr 709 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ V)
150 simpll 750 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)))
15184ad3antrrr 709 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ 𝒫 𝐶)
152109adantr 466 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ Fin)
153151, 152elind 3949 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
154 simpr 471 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘𝑦)
155121eqcomd 2777 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 = (𝐹 “ (𝐹𝑦)))
156155adantr 466 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑦 = (𝐹 “ (𝐹𝑦)))
157154, 156eleqtrd 2852 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
158157adantllr 698 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
159150, 153, 158jca31 504 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
160130anbi2d 614 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
161 imaeq2 5602 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
162161eleq2d 2836 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (𝑘 ∈ (𝐹𝑥) ↔ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
163160, 162anbi12d 616 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) ↔ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦)))))
164163imbi1d 330 . . . . . . . . . 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)))
165 rge0ssre 12486 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
166 ax-resscn 10198 . . . . . . . . . . . . 13 ℝ ⊆ ℂ
167165, 166sstri 3761 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℂ
168 simplll 758 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝜑)
169 simpllr 760 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
170 fimass 6222 . . . . . . . . . . . . . . . . 17 (𝐹:𝐶𝐴 → (𝐹𝑥) ⊆ 𝐴)
17123, 170syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹𝑥) ⊆ 𝐴)
172171ad2antrr 705 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → (𝐹𝑥) ⊆ 𝐴)
173 simpr 471 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘 ∈ (𝐹𝑥))
174172, 173sseldd 3753 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
175174adantllr 698 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
176 foelrni 6388 . . . . . . . . . . . . . . . 16 ((𝐹:𝐶onto𝐴𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
1774, 176sylan 569 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
178177adantlr 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
179 nfv 1995 . . . . . . . . . . . . . . . 16 𝑛 𝑘𝐴
180106, 179nfan 1980 . . . . . . . . . . . . . . 15 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴)
181 nfv 1995 . . . . . . . . . . . . . . 15 𝑛 𝐵 ∈ (0[,)+∞)
182 csbid 3690 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 / 𝑘𝐵 = 𝐵
183182eqcomi 2780 . . . . . . . . . . . . . . . . . . . . 21 𝐵 = 𝑘 / 𝑘𝐵
184183a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝑘 / 𝑘𝐵)
185 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) = 𝑘 → (𝐹𝑛) = 𝑘)
186185eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑛) = 𝑘𝑘 = (𝐹𝑛))
187186csbeq1d 3689 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑛) = 𝑘𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
1881873ad2ant3 1129 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
18938idi 2 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
1901893adant3 1126 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
191184, 188, 1903eqtrd 2809 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
1921913adant1r 1187 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
193 0xr 10291 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ ℝ*
194193a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 0 ∈ ℝ*)
195 pnfxr 10297 . . . . . . . . . . . . . . . . . . . . . . . . 25 +∞ ∈ ℝ*
196195a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ℝ*)
19764adantr 466 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ (0[,]+∞))
198 simpr 471 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ 𝐷 ∈ (0[,)+∞))
199194, 196, 197, 198eliccnelico 40271 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 = +∞)
200199eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ = 𝐷)
201 simpr 471 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝑛𝐶)
20264idi 2 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
20314elrnmpt1 5511 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝐶𝐷 ∈ (0[,]+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
204201, 202, 203syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛𝐶) → 𝐷 ∈ ran (𝑛𝐶𝐷))
205204adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
206200, 205eqeltrd 2850 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
207206adantllr 698 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
208 simpllr 760 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
209207, 208condan 819 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) → 𝐷 ∈ (0[,)+∞))
2102093adant3 1126 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐷 ∈ (0[,)+∞))
211192, 210eqeltrd 2850 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 ∈ (0[,)+∞))
2122113exp 1112 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
213212adantr 466 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
214180, 181, 213rexlimd 3174 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (∃𝑛𝐶 (𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞)))
215178, 214mpd 15 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → 𝐵 ∈ (0[,)+∞))
216168, 169, 175, 215syl21anc 1475 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ (0[,)+∞))
217167, 216sseldi 3750 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
218217idi 2 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
219164, 218vtoclg 3417 . . . . . . . . 9 ((𝐹𝑦) ∈ V → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))) → 𝐵 ∈ ℂ))
220149, 159, 219sylc 65 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
221100, 108, 36, 109, 124, 148, 220fsumf1of 40321 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
222 sumeq1 14626 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → Σ𝑛𝑥 𝐷 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
223222eqeq2d 2781 . . . . . . . 8 (𝑥 = (𝐹𝑦) → (Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷 ↔ Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷))
224223rspcev 3460 . . . . . . 7 (((𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22596, 221, 224syl2anc 573 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22671, 225rnmptssrn 39887 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
227 sumex 14625 . . . . . . 7 Σ𝑛𝑥 𝐷 ∈ V
228227a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 ∈ V)
2296, 171ssexd 4940 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑥) ∈ V)
230 elpwg 4306 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
231229, 230syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
232171, 231mpbird 247 . . . . . . . . . 10 (𝜑 → (𝐹𝑥) ∈ 𝒫 𝐴)
233232adantr 466 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ 𝒫 𝐴)
23423ffund 6188 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
235234adantr 466 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Fun 𝐹)
236 elinel2 3951 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ∈ Fin)
237236adantl 467 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
238 imafi 8418 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ Fin) → (𝐹𝑥) ∈ Fin)
239235, 237, 238syl2anc 573 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ Fin)
240233, 239elind 3949 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
241240adantlr 694 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
242 nfv 1995 . . . . . . . . . 10 𝑘 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
24398, 242nfan 1980 . . . . . . . . 9 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
244 nfv 1995 . . . . . . . . . 10 𝑛 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
245106, 244nfan 1980 . . . . . . . . 9 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
246236adantl 467 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
247111adantr 466 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
248 f1ores 6293 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴𝑥𝐶) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
249247, 142, 248syl2anc 573 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
250249adantlr 694 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
251145adantllr 698 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
252243, 245, 36, 246, 250, 251, 217fsumf1of 40321 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑘 ∈ (𝐹𝑥)𝐵 = Σ𝑛𝑥 𝐷)
253252eqcomd 2777 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
254 sumeq1 14626 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → Σ𝑘𝑦 𝐵 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
255254eqeq2d 2781 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵 ↔ Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵))
256255rspcev 3460 . . . . . . 7 (((𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
257241, 253, 256syl2anc 573 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
258228, 257rnmptssrn 39887 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷) ⊆ ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵))
259226, 258eqssd 3769 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) = ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
260259supeq1d 8511 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
2616adantr 466 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
26298, 261, 215sge0revalmpt 41109 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ))
2631adantr 466 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
264106, 263, 209sge0revalmpt 41109 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
265260, 262, 2643eqtr4d 2815 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
26669, 265pm2.61dan 814 1 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wnf 1856  wcel 2145  wrex 3062  Vcvv 3351  csb 3682  cin 3722  wss 3723  𝒫 cpw 4298  cmpt 4864  ccnv 5249  dom cdm 5250  ran crn 5251  cres 5252  cima 5253  Fun wfun 6024  wf 6026  1-1wf1 6027  ontowfo 6028  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6795  Fincfn 8112  supcsup 8505  cc 10139  cr 10140  0cc0 10141  +∞cpnf 10276  *cxr 10278   < clt 10279  [,)cico 12381  [,]cicc 12382  Σcsu 14623  Σ^csumge0 41093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-inf2 8705  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-er 7899  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-sup 8507  df-oi 8574  df-card 8968  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-div 10890  df-nn 11226  df-2 11284  df-3 11285  df-n0 11499  df-z 11584  df-uz 11893  df-rp 12035  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624  df-sumge0 41094
This theorem is referenced by:  sge0resrnlem  41134  sge0fodjrnlem  41147  sge0xp  41160  meadjiunlem  41196  isomenndlem  41261  ovnsubaddlem1  41301
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