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 40917
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 6182 . . . . . . 7 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶onto𝐴)
42, 3syl 17 . . . . . 6 (𝜑𝐹:𝐶onto𝐴)
5 fornex 7177 . . . . . 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[,]+∞))
10 eqid 2651 . . . . . 6 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
118, 9, 10fmptdf 6427 . . . . 5 (𝜑 → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
1211adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
13 pnfex 10131 . . . . . . . 8 +∞ ∈ V
14 eqid 2651 . . . . . . . . 9 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
1514elrnmpt 5404 . . . . . . . 8 (+∞ ∈ V → (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷))
1613, 15ax-mp 5 . . . . . . 7 (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷)
1716biimpi 206 . . . . . 6 (+∞ ∈ ran (𝑛𝐶𝐷) → ∃𝑛𝐶 +∞ = 𝐷)
1817adantl 481 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → ∃𝑛𝐶 +∞ = 𝐷)
19 sge0f1o.2 . . . . . . 7 𝑛𝜑
20 nfv 1883 . . . . . . 7 𝑛+∞ ∈ ran (𝑘𝐴𝐵)
21 simp3 1083 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = 𝐷)
22 f1of 6175 . . . . . . . . . . . . . . 15 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
232, 22syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶𝐴)
2423ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
25 sge0f1o.6 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
26 nfcv 2793 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛)
27 nfv 1883 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) = 𝐺
2826nfcsb1 3581 . . . . . . . . . . . . . . . 16 𝑘(𝐹𝑛) / 𝑘𝐵
29 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑘𝐷
3028, 29nfeq 2805 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) / 𝑘𝐵 = 𝐷
3127, 30nfim 1865 . . . . . . . . . . . . . 14 𝑘((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)
32 eqeq1 2655 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘 = 𝐺 ↔ (𝐹𝑛) = 𝐺))
33 csbeq1a 3575 . . . . . . . . . . . . . . . 16 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
3433eqeq1d 2653 . . . . . . . . . . . . . . 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 3295 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ 𝐴 → ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷))
3824, 25, 37sylc 65 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
3938eqcomd 2657 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
40393adant3 1101 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
4121, 40eqtrd 2685 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = (𝐹𝑛) / 𝑘𝐵)
42 simpl 472 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → 𝜑)
4342, 24jca 553 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝜑 ∧ (𝐹𝑛) ∈ 𝐴))
44 nfv 1883 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) ∈ 𝐴
458, 44nfan 1868 . . . . . . . . . . . . . 14 𝑘(𝜑 ∧ (𝐹𝑛) ∈ 𝐴)
4628nfel1 2808 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)
4745, 46nfim 1865 . . . . . . . . . . . . 13 𝑘((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
48 eleq1 2718 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘𝐴 ↔ (𝐹𝑛) ∈ 𝐴))
4948anbi2d 740 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → ((𝜑𝑘𝐴) ↔ (𝜑 ∧ (𝐹𝑛) ∈ 𝐴)))
5033eleq1d 2715 . . . . . . . . . . . . . 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 3295 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ 𝐴 → ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
5324, 43, 52sylc 65 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
5428, 10, 33elrnmpt1sf 39690 . . . . . . . . . . 11 (((𝐹𝑛) ∈ 𝐴(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5524, 53, 54syl2anc 694 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
56553adant3 1101 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5741, 56eqeltrd 2730 . . . . . . . 8 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ ∈ ran (𝑘𝐴𝐵))
58573exp 1283 . . . . . . 7 (𝜑 → (𝑛𝐶 → (+∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵))))
5919, 20, 58rexlimd 3055 . . . . . 6 (𝜑 → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6059adantr 480 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6118, 60mpd 15 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑘𝐴𝐵))
627, 12, 61sge0pnfval 40908 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = +∞)
631adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
6439, 53eqeltrd 2730 . . . . . 6 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
6519, 64, 14fmptdf 6427 . . . . 5 (𝜑 → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
6665adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
67 simpr 476 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑛𝐶𝐷))
6863, 66, 67sge0pnfval 40908 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = +∞)
6962, 68eqtr4d 2688 . 2 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
70 sumex 14462 . . . . . . 7 Σ𝑘𝑦 𝐵 ∈ V
7170a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 ∈ V)
72 cnvimass 5520 . . . . . . . . . . . . 13 (𝐹𝑦) ⊆ dom 𝐹
7372a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑦) ⊆ dom 𝐹)
74 fdm 6089 . . . . . . . . . . . . 13 (𝐹:𝐶𝐴 → dom 𝐹 = 𝐶)
7523, 74syl 17 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐶)
7673, 75sseqtrd 3674 . . . . . . . . . . 11 (𝜑 → (𝐹𝑦) ⊆ 𝐶)
77 fex 6530 . . . . . . . . . . . . . . 15 ((𝐹:𝐶𝐴𝐶𝑉) → 𝐹 ∈ V)
7823, 1, 77syl2anc 694 . . . . . . . . . . . . . 14 (𝜑𝐹 ∈ V)
79 cnvexg 7154 . . . . . . . . . . . . . 14 (𝐹 ∈ V → 𝐹 ∈ V)
8078, 79syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
81 imaexg 7145 . . . . . . . . . . . . 13 (𝐹 ∈ V → (𝐹𝑦) ∈ V)
8280, 81syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑦) ∈ V)
83 elpwg 4199 . . . . . . . . . . . 12 ((𝐹𝑦) ∈ V → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8482, 83syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8576, 84mpbird 247 . . . . . . . . . 10 (𝜑 → (𝐹𝑦) ∈ 𝒫 𝐶)
8685adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ 𝒫 𝐶)
87 f1ocnv 6187 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐴1-1-onto𝐶)
882, 87syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐴1-1-onto𝐶)
89 f1ofun 6177 . . . . . . . . . . . 12 (𝐹:𝐴1-1-onto𝐶 → Fun 𝐹)
9088, 89syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
9190adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Fun 𝐹)
92 elinel2 3833 . . . . . . . . . . 11 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
9392adantl 481 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
94 imafi 8300 . . . . . . . . . 10 ((Fun 𝐹𝑦 ∈ Fin) → (𝐹𝑦) ∈ Fin)
9591, 93, 94syl2anc 694 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
9686, 95elind 3831 . . . . . . . 8 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
9796adantlr 751 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
98 nfv 1883 . . . . . . . . . 10 𝑘 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
998, 98nfan 1868 . . . . . . . . 9 𝑘(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
100 nfv 1883 . . . . . . . . 9 𝑘 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
10199, 100nfan 1868 . . . . . . . 8 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
102 nfcv 2793 . . . . . . . . . . . 12 𝑛+∞
103 nfmpt1 4780 . . . . . . . . . . . . 13 𝑛(𝑛𝐶𝐷)
104103nfrn 5400 . . . . . . . . . . . 12 𝑛ran (𝑛𝐶𝐷)
105102, 104nfel 2806 . . . . . . . . . . 11 𝑛+∞ ∈ ran (𝑛𝐶𝐷)
106105nfn 1824 . . . . . . . . . 10 𝑛 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
10719, 106nfan 1868 . . . . . . . . 9 𝑛(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
108 nfv 1883 . . . . . . . . 9 𝑛 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
109107, 108nfan 1868 . . . . . . . 8 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
11095adantlr 751 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
111 f1of1 6174 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1𝐴)
1122, 111syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐶1-1𝐴)
113112adantr 480 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
11484adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
11586, 114mpbid 222 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ⊆ 𝐶)
116 f1ores 6189 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴 ∧ (𝐹𝑦) ⊆ 𝐶) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
117113, 115, 116syl2anc 694 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
1184adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶onto𝐴)
119 elpwinss 39530 . . . . . . . . . . . . 13 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦𝐴)
120119adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦𝐴)
121 foimacnv 6192 . . . . . . . . . . . 12 ((𝐹:𝐶onto𝐴𝑦𝐴) → (𝐹 “ (𝐹𝑦)) = 𝑦)
122118, 120, 121syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
123122f1oeq3d 6172 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)) ↔ (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦))
124117, 123mpbid 222 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
125124adantlr 751 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
12682ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ V)
127 simpll 805 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝜑)
12896adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
129 simpr 476 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐹𝑦))
130127, 128, 129jca31 556 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)))
131 eleq1 2718 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)))
132131anbi2d 740 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ (𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
133 eleq2 2719 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝑛𝑥𝑛 ∈ (𝐹𝑦)))
134132, 133anbi12d 747 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) ↔ ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦))))
135 reseq2 5423 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 ↾ (𝐹𝑦)))
136135fveq1d 6231 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝐹𝑥)‘𝑛) = ((𝐹 ↾ (𝐹𝑦))‘𝑛))
137136eqeq1d 2653 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝐹𝑥)‘𝑛) = 𝐺 ↔ ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
138134, 137imbi12d 333 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺) ↔ (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)))
139 fvres 6245 . . . . . . . . . . . . 13 (𝑛𝑥 → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
140139adantl 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
141 simpll 805 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝜑)
142 elpwinss 39530 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥𝐶)
143142adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥𝐶)
144143sselda 3636 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝑛𝐶)
145141, 144, 25syl2anc 694 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → (𝐹𝑛) = 𝐺)
146140, 145eqtrd 2685 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
147138, 146vtoclg 3297 . . . . . . . . . 10 ((𝐹𝑦) ∈ V → (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
148126, 130, 147sylc 65 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
149148adantllr 755 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
15082ad3antrrr 766 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ V)
151 simpll 805 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)))
15285ad3antrrr 766 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ 𝒫 𝐶)
153110adantr 480 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ Fin)
154152, 153elind 3831 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
155 simpr 476 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘𝑦)
156122eqcomd 2657 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 = (𝐹 “ (𝐹𝑦)))
157156adantr 480 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑦 = (𝐹 “ (𝐹𝑦)))
158155, 157eleqtrd 2732 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
159158adantllr 755 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
160151, 154, 159jca31 556 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
161131anbi2d 740 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
162 imaeq2 5497 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
163162eleq2d 2716 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (𝑘 ∈ (𝐹𝑥) ↔ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
164161, 163anbi12d 747 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) ↔ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦)))))
165164imbi1d 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)))
166 rge0ssre 12318 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
167 ax-resscn 10031 . . . . . . . . . . . . 13 ℝ ⊆ ℂ
168166, 167sstri 3645 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℂ
169 simplll 813 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝜑)
170 simpllr 815 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
171 fimass 6119 . . . . . . . . . . . . . . . . 17 (𝐹:𝐶𝐴 → (𝐹𝑥) ⊆ 𝐴)
17223, 171syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹𝑥) ⊆ 𝐴)
173172ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → (𝐹𝑥) ⊆ 𝐴)
174 simpr 476 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘 ∈ (𝐹𝑥))
175173, 174sseldd 3637 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
176175adantllr 755 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
177 foelrni 6283 . . . . . . . . . . . . . . . 16 ((𝐹:𝐶onto𝐴𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
1784, 177sylan 487 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
179178adantlr 751 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
180 nfv 1883 . . . . . . . . . . . . . . . 16 𝑛 𝑘𝐴
181107, 180nfan 1868 . . . . . . . . . . . . . . 15 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴)
182 nfv 1883 . . . . . . . . . . . . . . 15 𝑛 𝐵 ∈ (0[,)+∞)
183 csbid 3574 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 / 𝑘𝐵 = 𝐵
184183eqcomi 2660 . . . . . . . . . . . . . . . . . . . . 21 𝐵 = 𝑘 / 𝑘𝐵
185184a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝑘 / 𝑘𝐵)
186 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) = 𝑘 → (𝐹𝑛) = 𝑘)
187186eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑛) = 𝑘𝑘 = (𝐹𝑛))
188187csbeq1d 3573 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑛) = 𝑘𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
1891883ad2ant3 1104 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
19038idi 2 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
1911903adant3 1101 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
192185, 189, 1913eqtrd 2689 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
1931923adant1r 1359 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
194 0xr 10124 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ ℝ*
195194a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 0 ∈ ℝ*)
196 pnfxr 10130 . . . . . . . . . . . . . . . . . . . . . . . . 25 +∞ ∈ ℝ*
197196a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ℝ*)
19864adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ (0[,]+∞))
199 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ 𝐷 ∈ (0[,)+∞))
200195, 197, 198, 199eliccnelico 40074 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 = +∞)
201200eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ = 𝐷)
202 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝑛𝐶)
20364idi 2 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
20414elrnmpt1 5406 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝐶𝐷 ∈ (0[,]+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
205202, 203, 204syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛𝐶) → 𝐷 ∈ ran (𝑛𝐶𝐷))
206205adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
207201, 206eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
208207adantllr 755 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
209 simpllr 815 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
210208, 209condan 852 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) → 𝐷 ∈ (0[,)+∞))
2112103adant3 1101 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐷 ∈ (0[,)+∞))
212193, 211eqeltrd 2730 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 ∈ (0[,)+∞))
2132123exp 1283 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
214213adantr 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
215181, 182, 214rexlimd 3055 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (∃𝑛𝐶 (𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞)))
216179, 215mpd 15 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → 𝐵 ∈ (0[,)+∞))
217169, 170, 176, 216syl21anc 1365 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ (0[,)+∞))
218168, 217sseldi 3634 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
219218idi 2 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
220165, 219vtoclg 3297 . . . . . . . . 9 ((𝐹𝑦) ∈ V → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))) → 𝐵 ∈ ℂ))
221150, 160, 220sylc 65 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
222101, 109, 36, 110, 125, 149, 221fsumf1of 40124 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
223 sumeq1 14463 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → Σ𝑛𝑥 𝐷 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
224223eqeq2d 2661 . . . . . . . 8 (𝑥 = (𝐹𝑦) → (Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷 ↔ Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷))
225224rspcev 3340 . . . . . . 7 (((𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22697, 222, 225syl2anc 694 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22771, 226rnmptssrn 39682 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
228 sumex 14462 . . . . . . 7 Σ𝑛𝑥 𝐷 ∈ V
229228a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 ∈ V)
2306, 172ssexd 4838 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑥) ∈ V)
231 elpwg 4199 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
232230, 231syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
233172, 232mpbird 247 . . . . . . . . . 10 (𝜑 → (𝐹𝑥) ∈ 𝒫 𝐴)
234233adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ 𝒫 𝐴)
235 ffun 6086 . . . . . . . . . . . 12 (𝐹:𝐶𝐴 → Fun 𝐹)
23623, 235syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
237236adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Fun 𝐹)
238 elinel2 3833 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ∈ Fin)
239238adantl 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
240 imafi 8300 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ Fin) → (𝐹𝑥) ∈ Fin)
241237, 239, 240syl2anc 694 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ Fin)
242234, 241elind 3831 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
243242adantlr 751 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
244 nfv 1883 . . . . . . . . . 10 𝑘 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
24599, 244nfan 1868 . . . . . . . . 9 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
246 nfv 1883 . . . . . . . . . 10 𝑛 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
247107, 246nfan 1868 . . . . . . . . 9 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
248238adantl 481 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
249112adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
250 f1ores 6189 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴𝑥𝐶) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
251249, 143, 250syl2anc 694 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
252251adantlr 751 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
253146adantllr 755 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
254245, 247, 36, 248, 252, 253, 218fsumf1of 40124 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑘 ∈ (𝐹𝑥)𝐵 = Σ𝑛𝑥 𝐷)
255254eqcomd 2657 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
256 sumeq1 14463 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → Σ𝑘𝑦 𝐵 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
257256eqeq2d 2661 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵 ↔ Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵))
258257rspcev 3340 . . . . . . 7 (((𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
259243, 255, 258syl2anc 694 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
260229, 259rnmptssrn 39682 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷) ⊆ ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵))
261227, 260eqssd 3653 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) = ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
262261supeq1d 8393 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
2636adantr 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
26499, 263, 216sge0revalmpt 40913 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ))
2651adantr 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
266107, 265, 210sge0revalmpt 40913 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
267262, 264, 2663eqtr4d 2695 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
26869, 267pm2.61dan 849 1 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wrex 2942  Vcvv 3231  csb 3566  cin 3606  wss 3607  𝒫 cpw 4191  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  Fincfn 7997  supcsup 8387  cc 9972  cr 9973  0cc0 9974  +∞cpnf 10109  *cxr 10111   < clt 10112  [,)cico 12215  [,]cicc 12216  Σcsu 14460  Σ^csumge0 40897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-sumge0 40898
This theorem is referenced by:  sge0resrnlem  40938  sge0fodjrnlem  40951  sge0xp  40964  meadjiunlem  41000  isomenndlem  41065  ovnsubaddlem1  41105
  Copyright terms: Public domain W3C validator