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Theorem sge0f1o 45908
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 6845 . . . . . . 7 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶onto𝐴)
42, 3syl 17 . . . . . 6 (𝜑𝐹:𝐶onto𝐴)
5 focdmex 7960 . . . . . 6 (𝐶𝑉 → (𝐹:𝐶onto𝐴𝐴 ∈ V))
61, 4, 5sylc 65 . . . . 5 (𝜑𝐴 ∈ V)
76adantr 479 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
8 sge0f1o.1 . . . . . 6 𝑘𝜑
9 sge0f1o.7 . . . . . 6 ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
10 eqid 2725 . . . . . 6 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
118, 9, 10fmptdf 7126 . . . . 5 (𝜑 → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
1211adantr 479 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑘𝐴𝐵):𝐴⟶(0[,]+∞))
13 pnfex 11299 . . . . . . . 8 +∞ ∈ V
14 eqid 2725 . . . . . . . . 9 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
1514elrnmpt 5958 . . . . . . . 8 (+∞ ∈ V → (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷))
1613, 15ax-mp 5 . . . . . . 7 (+∞ ∈ ran (𝑛𝐶𝐷) ↔ ∃𝑛𝐶 +∞ = 𝐷)
1716biimpi 215 . . . . . 6 (+∞ ∈ ran (𝑛𝐶𝐷) → ∃𝑛𝐶 +∞ = 𝐷)
1817adantl 480 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → ∃𝑛𝐶 +∞ = 𝐷)
19 sge0f1o.2 . . . . . . 7 𝑛𝜑
20 nfv 1909 . . . . . . 7 𝑛+∞ ∈ ran (𝑘𝐴𝐵)
21 simp3 1135 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = 𝐷)
22 f1of 6838 . . . . . . . . . . . . . . 15 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
232, 22syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶𝐴)
2423ffvelcdmda 7093 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
25 sge0f1o.6 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
26 nfcv 2891 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛)
27 nfv 1909 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) = 𝐺
2826nfcsb1 3913 . . . . . . . . . . . . . . . 16 𝑘(𝐹𝑛) / 𝑘𝐵
29 nfcv 2891 . . . . . . . . . . . . . . . 16 𝑘𝐷
3028, 29nfeq 2905 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) / 𝑘𝐵 = 𝐷
3127, 30nfim 1891 . . . . . . . . . . . . . 14 𝑘((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)
32 eqeq1 2729 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘 = 𝐺 ↔ (𝐹𝑛) = 𝐺))
33 csbeq1a 3903 . . . . . . . . . . . . . . . 16 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
3433eqeq1d 2727 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝐵 = 𝐷(𝐹𝑛) / 𝑘𝐵 = 𝐷))
3532, 34imbi12d 343 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → ((𝑘 = 𝐺𝐵 = 𝐷) ↔ ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷)))
36 sge0f1o.3 . . . . . . . . . . . . . 14 (𝑘 = 𝐺𝐵 = 𝐷)
3726, 31, 35, 36vtoclgf 3548 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ 𝐴 → ((𝐹𝑛) = 𝐺(𝐹𝑛) / 𝑘𝐵 = 𝐷))
3824, 25, 37sylc 65 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
3938eqcomd 2731 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
40393adant3 1129 . . . . . . . . . 10 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → 𝐷 = (𝐹𝑛) / 𝑘𝐵)
4121, 40eqtrd 2765 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ = (𝐹𝑛) / 𝑘𝐵)
42 simpl 481 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → 𝜑)
4342, 24jca 510 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → (𝜑 ∧ (𝐹𝑛) ∈ 𝐴))
44 nfv 1909 . . . . . . . . . . . . . . 15 𝑘(𝐹𝑛) ∈ 𝐴
458, 44nfan 1894 . . . . . . . . . . . . . 14 𝑘(𝜑 ∧ (𝐹𝑛) ∈ 𝐴)
4628nfel1 2908 . . . . . . . . . . . . . 14 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)
4745, 46nfim 1891 . . . . . . . . . . . . 13 𝑘((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
48 eleq1 2813 . . . . . . . . . . . . . . 15 (𝑘 = (𝐹𝑛) → (𝑘𝐴 ↔ (𝐹𝑛) ∈ 𝐴))
4948anbi2d 628 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → ((𝜑𝑘𝐴) ↔ (𝜑 ∧ (𝐹𝑛) ∈ 𝐴)))
5033eleq1d 2810 . . . . . . . . . . . . . 14 (𝑘 = (𝐹𝑛) → (𝐵 ∈ (0[,]+∞) ↔ (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
5149, 50imbi12d 343 . . . . . . . . . . . . 13 (𝑘 = (𝐹𝑛) → (((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))))
5226, 47, 51, 9vtoclgf 3548 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ 𝐴 → ((𝜑 ∧ (𝐹𝑛) ∈ 𝐴) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)))
5324, 43, 52sylc 65 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞))
5428, 10, 33elrnmpt1sf 44701 . . . . . . . . . . 11 (((𝐹𝑛) ∈ 𝐴(𝐹𝑛) / 𝑘𝐵 ∈ (0[,]+∞)) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5524, 53, 54syl2anc 582 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
56553adant3 1129 . . . . . . . . 9 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → (𝐹𝑛) / 𝑘𝐵 ∈ ran (𝑘𝐴𝐵))
5741, 56eqeltrd 2825 . . . . . . . 8 ((𝜑𝑛𝐶 ∧ +∞ = 𝐷) → +∞ ∈ ran (𝑘𝐴𝐵))
58573exp 1116 . . . . . . 7 (𝜑 → (𝑛𝐶 → (+∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵))))
5919, 20, 58rexlimd 3253 . . . . . 6 (𝜑 → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6059adantr 479 . . . . 5 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (∃𝑛𝐶 +∞ = 𝐷 → +∞ ∈ ran (𝑘𝐴𝐵)))
6118, 60mpd 15 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑘𝐴𝐵))
627, 12, 61sge0pnfval 45899 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = +∞)
631adantr 479 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
6439, 53eqeltrd 2825 . . . . . 6 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
6519, 64, 14fmptdf 7126 . . . . 5 (𝜑 → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
6665adantr 479 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶𝐷):𝐶⟶(0[,]+∞))
67 simpr 483 . . . 4 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → +∞ ∈ ran (𝑛𝐶𝐷))
6863, 66, 67sge0pnfval 45899 . . 3 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = +∞)
6962, 68eqtr4d 2768 . 2 ((𝜑 ∧ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
70 sumex 15670 . . . . . . 7 Σ𝑘𝑦 𝐵 ∈ V
7170a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 ∈ V)
72 cnvimass 6086 . . . . . . . . . . . 12 (𝐹𝑦) ⊆ dom 𝐹
7372, 23fssdm 6742 . . . . . . . . . . 11 (𝜑 → (𝐹𝑦) ⊆ 𝐶)
7423, 1fexd 7239 . . . . . . . . . . . . . 14 (𝜑𝐹 ∈ V)
75 cnvexg 7932 . . . . . . . . . . . . . 14 (𝐹 ∈ V → 𝐹 ∈ V)
7674, 75syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
77 imaexg 7921 . . . . . . . . . . . . 13 (𝐹 ∈ V → (𝐹𝑦) ∈ V)
7876, 77syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑦) ∈ V)
79 elpwg 4607 . . . . . . . . . . . 12 ((𝐹𝑦) ∈ V → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8078, 79syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
8173, 80mpbird 256 . . . . . . . . . 10 (𝜑 → (𝐹𝑦) ∈ 𝒫 𝐶)
8281adantr 479 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ 𝒫 𝐶)
83 f1ocnv 6850 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐴1-1-onto𝐶)
842, 83syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐴1-1-onto𝐶)
85 f1ofun 6840 . . . . . . . . . . . 12 (𝐹:𝐴1-1-onto𝐶 → Fun 𝐹)
8684, 85syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
8786adantr 479 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Fun 𝐹)
88 elinel2 4194 . . . . . . . . . . 11 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
8988adantl 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
90 imafi 9200 . . . . . . . . . 10 ((Fun 𝐹𝑦 ∈ Fin) → (𝐹𝑦) ∈ Fin)
9187, 89, 90syl2anc 582 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
9282, 91elind 4192 . . . . . . . 8 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
9392adantlr 713 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
94 nfv 1909 . . . . . . . . . 10 𝑘 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
958, 94nfan 1894 . . . . . . . . 9 𝑘(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
96 nfv 1909 . . . . . . . . 9 𝑘 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
9795, 96nfan 1894 . . . . . . . 8 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
98 nfcv 2891 . . . . . . . . . . . 12 𝑛+∞
99 nfmpt1 5257 . . . . . . . . . . . . 13 𝑛(𝑛𝐶𝐷)
10099nfrn 5954 . . . . . . . . . . . 12 𝑛ran (𝑛𝐶𝐷)
10198, 100nfel 2906 . . . . . . . . . . 11 𝑛+∞ ∈ ran (𝑛𝐶𝐷)
102101nfn 1852 . . . . . . . . . 10 𝑛 ¬ +∞ ∈ ran (𝑛𝐶𝐷)
10319, 102nfan 1894 . . . . . . . . 9 𝑛(𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷))
104 nfv 1909 . . . . . . . . 9 𝑛 𝑦 ∈ (𝒫 𝐴 ∩ Fin)
105103, 104nfan 1894 . . . . . . . 8 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
10691adantlr 713 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ∈ Fin)
107 f1of1 6837 . . . . . . . . . . . . 13 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1𝐴)
1082, 107syl 17 . . . . . . . . . . . 12 (𝜑𝐹:𝐶1-1𝐴)
109108adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
11080adantr 479 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹𝑦) ∈ 𝒫 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
11182, 110mpbid 231 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹𝑦) ⊆ 𝐶)
112 f1ores 6852 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴 ∧ (𝐹𝑦) ⊆ 𝐶) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
113109, 111, 112syl2anc 582 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)))
1144adantr 479 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐶onto𝐴)
115 elpwinss 44555 . . . . . . . . . . . . 13 (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦𝐴)
116115adantl 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦𝐴)
117 foimacnv 6855 . . . . . . . . . . . 12 ((𝐹:𝐶onto𝐴𝑦𝐴) → (𝐹 “ (𝐹𝑦)) = 𝑦)
118114, 116, 117syl2anc 582 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
119118f1oeq3d 6835 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto→(𝐹 “ (𝐹𝑦)) ↔ (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦))
120113, 119mpbid 231 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
121120adantlr 713 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ (𝐹𝑦)):(𝐹𝑦)–1-1-onto𝑦)
12278ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ V)
123 simpll 765 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝜑)
12492adantr 479 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
125 simpr 483 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐹𝑦))
126123, 124, 125jca31 513 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)))
127 eleq1 2813 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)))
128127anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ (𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
129 eleq2 2814 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝑛𝑥𝑛 ∈ (𝐹𝑦)))
130128, 129anbi12d 630 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) ↔ ((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦))))
131 reseq2 5980 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 ↾ (𝐹𝑦)))
132131fveq1d 6898 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → ((𝐹𝑥)‘𝑛) = ((𝐹 ↾ (𝐹𝑦))‘𝑛))
133132eqeq1d 2727 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝐹𝑥)‘𝑛) = 𝐺 ↔ ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
134130, 133imbi12d 343 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺) ↔ (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)))
135 fvres 6915 . . . . . . . . . . . . 13 (𝑛𝑥 → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
136135adantl 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = (𝐹𝑛))
137 simpll 765 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝜑)
138 elpwinss 44555 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥𝐶)
139138adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥𝐶)
140139sselda 3976 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → 𝑛𝐶)
141137, 140, 25syl2anc 582 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → (𝐹𝑛) = 𝐺)
142136, 141eqtrd 2765 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
143134, 142vtoclg 3532 . . . . . . . . . 10 ((𝐹𝑦) ∈ V → (((𝜑 ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺))
144122, 126, 143sylc 65 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
145144adantllr 717 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑛 ∈ (𝐹𝑦)) → ((𝐹 ↾ (𝐹𝑦))‘𝑛) = 𝐺)
14678ad3antrrr 728 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ V)
147 simpll 765 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)))
14881ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ 𝒫 𝐶)
149106adantr 479 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ Fin)
150148, 149elind 4192 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))
151 simpr 483 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘𝑦)
152118eqcomd 2731 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 = (𝐹 “ (𝐹𝑦)))
153152adantr 479 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑦 = (𝐹 “ (𝐹𝑦)))
154151, 153eleqtrd 2827 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
155154adantllr 717 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝑘 ∈ (𝐹 “ (𝐹𝑦)))
156147, 150, 155jca31 513 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
157127anbi2d 628 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ↔ ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin))))
158 imaeq2 6060 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
159158eleq2d 2811 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑦) → (𝑘 ∈ (𝐹𝑥) ↔ 𝑘 ∈ (𝐹 “ (𝐹𝑦))))
160157, 159anbi12d 630 . . . . . . . . . . 11 (𝑥 = (𝐹𝑦) → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) ↔ (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦)))))
161160imbi1d 340 . . . . . . . . . 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)))
162 rge0ssre 13468 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
163 ax-resscn 11197 . . . . . . . . . . . . 13 ℝ ⊆ ℂ
164162, 163sstri 3986 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℂ
165 simplll 773 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝜑)
166 simpllr 774 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
167 fimass 6743 . . . . . . . . . . . . . . . . 17 (𝐹:𝐶𝐴 → (𝐹𝑥) ⊆ 𝐴)
16823, 167syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹𝑥) ⊆ 𝐴)
169168ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → (𝐹𝑥) ⊆ 𝐴)
170 simpr 483 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘 ∈ (𝐹𝑥))
171169, 170sseldd 3977 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
172171adantllr 717 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝑘𝐴)
173 foelcdmi 6959 . . . . . . . . . . . . . . . 16 ((𝐹:𝐶onto𝐴𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
1744, 173sylan 578 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
175174adantlr 713 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → ∃𝑛𝐶 (𝐹𝑛) = 𝑘)
176 nfv 1909 . . . . . . . . . . . . . . . 16 𝑛 𝑘𝐴
177103, 176nfan 1894 . . . . . . . . . . . . . . 15 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴)
178 nfv 1909 . . . . . . . . . . . . . . 15 𝑛 𝐵 ∈ (0[,)+∞)
179 csbid 3902 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 / 𝑘𝐵 = 𝐵
180179eqcomi 2734 . . . . . . . . . . . . . . . . . . . . 21 𝐵 = 𝑘 / 𝑘𝐵
181180a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝑘 / 𝑘𝐵)
182 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) = 𝑘 → (𝐹𝑛) = 𝑘)
183182eqcomd 2731 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑛) = 𝑘𝑘 = (𝐹𝑛))
184183csbeq1d 3893 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑛) = 𝑘𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
1851843ad2ant3 1132 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝑘 / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
18638idi 1 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛𝐶) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
1871863adant3 1129 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → (𝐹𝑛) / 𝑘𝐵 = 𝐷)
188181, 185, 1873eqtrd 2769 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
1891883adant1r 1174 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 = 𝐷)
190 0xr 11293 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ ℝ*
191190a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 0 ∈ ℝ*)
192 pnfxr 11300 . . . . . . . . . . . . . . . . . . . . . . . . 25 +∞ ∈ ℝ*
193192a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ℝ*)
19464adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ (0[,]+∞))
195 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ 𝐷 ∈ (0[,)+∞))
196191, 193, 194, 195eliccnelico 45052 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 = +∞)
197196eqcomd 2731 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ = 𝐷)
198 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝑛𝐶)
19964idi 1 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛𝐶) → 𝐷 ∈ (0[,]+∞))
20014elrnmpt1 5960 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝐶𝐷 ∈ (0[,]+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
201198, 199, 200syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛𝐶) → 𝐷 ∈ ran (𝑛𝐶𝐷))
202201adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → 𝐷 ∈ ran (𝑛𝐶𝐷))
203197, 202eqeltrd 2825 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
204203adantllr 717 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → +∞ ∈ ran (𝑛𝐶𝐷))
205 simpllr 774 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) ∧ ¬ 𝐷 ∈ (0[,)+∞)) → ¬ +∞ ∈ ran (𝑛𝐶𝐷))
206204, 205condan 816 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶) → 𝐷 ∈ (0[,)+∞))
2072063adant3 1129 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐷 ∈ (0[,)+∞))
208189, 207eqeltrd 2825 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑛𝐶 ∧ (𝐹𝑛) = 𝑘) → 𝐵 ∈ (0[,)+∞))
2092083exp 1116 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
210209adantr 479 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (𝑛𝐶 → ((𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞))))
211177, 178, 210rexlimd 3253 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → (∃𝑛𝐶 (𝐹𝑛) = 𝑘𝐵 ∈ (0[,)+∞)))
212175, 211mpd 15 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑘𝐴) → 𝐵 ∈ (0[,)+∞))
213165, 166, 172, 212syl21anc 836 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ (0[,)+∞))
214164, 213sselid 3974 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
215214idi 1 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹𝑥)) → 𝐵 ∈ ℂ)
216161, 215vtoclg 3532 . . . . . . . . 9 ((𝐹𝑦) ∈ V → ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ (𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (𝐹 “ (𝐹𝑦))) → 𝐵 ∈ ℂ))
217146, 156, 216sylc 65 . . . . . . . 8 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
21897, 105, 36, 106, 121, 145, 217fsumf1of 45100 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
219 sumeq1 15671 . . . . . . . 8 (𝑥 = (𝐹𝑦) → Σ𝑛𝑥 𝐷 = Σ𝑛 ∈ (𝐹𝑦)𝐷)
220219rspceeqv 3628 . . . . . . 7 (((𝐹𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ Σ𝑘𝑦 𝐵 = Σ𝑛 ∈ (𝐹𝑦)𝐷) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22193, 218, 220syl2anc 582 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)Σ𝑘𝑦 𝐵 = Σ𝑛𝑥 𝐷)
22271, 221rnmptssrn 44694 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
223 sumex 15670 . . . . . . 7 Σ𝑛𝑥 𝐷 ∈ V
224223a1i 11 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 ∈ V)
2256, 168ssexd 5325 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑥) ∈ V)
226 elpwg 4607 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
227225, 226syl 17 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑥) ∈ 𝒫 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
228168, 227mpbird 256 . . . . . . . . . 10 (𝜑 → (𝐹𝑥) ∈ 𝒫 𝐴)
229228adantr 479 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ 𝒫 𝐴)
23023ffund 6727 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
231230adantr 479 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Fun 𝐹)
232 elinel2 4194 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) → 𝑥 ∈ Fin)
233232adantl 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
234 imafi 9200 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ Fin) → (𝐹𝑥) ∈ Fin)
235231, 233, 234syl2anc 582 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ Fin)
236229, 235elind 4192 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
237236adantlr 713 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin))
238 nfv 1909 . . . . . . . . . 10 𝑘 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
23995, 238nfan 1894 . . . . . . . . 9 𝑘((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
240 nfv 1909 . . . . . . . . . 10 𝑛 𝑥 ∈ (𝒫 𝐶 ∩ Fin)
241103, 240nfan 1894 . . . . . . . . 9 𝑛((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin))
242232adantl 480 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑥 ∈ Fin)
243108adantr 479 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐶1-1𝐴)
244 f1ores 6852 . . . . . . . . . . 11 ((𝐹:𝐶1-1𝐴𝑥𝐶) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
245243, 139, 244syl2anc 582 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
246245adantlr 713 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹𝑥):𝑥1-1-onto→(𝐹𝑥))
247142adantllr 717 . . . . . . . . 9 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑛𝑥) → ((𝐹𝑥)‘𝑛) = 𝐺)
248239, 241, 36, 242, 246, 247, 214fsumf1of 45100 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑘 ∈ (𝐹𝑥)𝐵 = Σ𝑛𝑥 𝐷)
249248eqcomd 2731 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
250 sumeq1 15671 . . . . . . . 8 (𝑦 = (𝐹𝑥) → Σ𝑘𝑦 𝐵 = Σ𝑘 ∈ (𝐹𝑥)𝐵)
251250rspceeqv 3628 . . . . . . 7 (((𝐹𝑥) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑛𝑥 𝐷 = Σ𝑘 ∈ (𝐹𝑥)𝐵) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
252237, 249, 251syl2anc 582 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) ∧ 𝑥 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑛𝑥 𝐷 = Σ𝑘𝑦 𝐵)
253224, 252rnmptssrn 44694 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷) ⊆ ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵))
254222, 253eqssd 3994 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵) = ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷))
255254supeq1d 9471 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
2566adantr 479 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐴 ∈ V)
25795, 256, 212sge0revalmpt 45904 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑦 𝐵), ℝ*, < ))
2581adantr 479 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → 𝐶𝑉)
259103, 258, 206sge0revalmpt 45904 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑛𝐶𝐷)) = sup(ran (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↦ Σ𝑛𝑥 𝐷), ℝ*, < ))
260255, 257, 2593eqtr4d 2775 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran (𝑛𝐶𝐷)) → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
26169, 260pm2.61dan 811 1 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wnf 1777  wcel 2098  wrex 3059  Vcvv 3461  csb 3889  cin 3943  wss 3944  𝒫 cpw 4604  cmpt 5232  ccnv 5677  ran crn 5679  cres 5680  cima 5681  Fun wfun 6543  wf 6545  1-1wf1 6546  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  Fincfn 8964  supcsup 9465  cc 11138  cr 11139  0cc0 11140  +∞cpnf 11277  *cxr 11279   < clt 11280  [,)cico 13361  [,]cicc 13362  Σcsu 15668  Σ^csumge0 45888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-sum 15669  df-sumge0 45889
This theorem is referenced by:  sge0resrnlem  45929  sge0fodjrnlem  45942  sge0xp  45955  meadjiunlem  45991  isomenndlem  46056  ovnsubaddlem1  46096
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