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Theorem fsum3 11096
Description: The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
Hypotheses
Ref Expression
fsum.1 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
fsum.2 (𝜑𝑀 ∈ ℕ)
fsum.3 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
fsum.4 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fsum.5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
Assertion
Ref Expression
fsum3 (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑛   𝐶,𝑘   𝑘,𝐹,𝑛   𝑘,𝐺,𝑛   𝑘,𝑀,𝑛   𝜑,𝑘,𝑛
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑛)

Proof of Theorem fsum3
Dummy variables 𝑓 𝑖 𝑗 𝑚 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sumdc 11063 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
2 nnuz 9310 . . . . 5 ℕ = (ℤ‘1)
3 1zzd 9032 . . . . 5 (𝜑 → 1 ∈ ℤ)
4 elnnuz 9311 . . . . . 6 (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ‘1))
52eqimss2i 3122 . . . . . . . . . 10 (ℤ‘1) ⊆ ℕ
65sseli 3061 . . . . . . . . 9 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℕ)
76adantl 273 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℕ)
8 fveq2 5387 . . . . . . . . . . 11 (𝑛 = 𝑥 → (𝐺𝑛) = (𝐺𝑥))
98eleq1d 2184 . . . . . . . . . 10 (𝑛 = 𝑥 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑥) ∈ ℂ))
10 fsum.1 . . . . . . . . . . . 12 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
11 fsum.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℕ)
12 fsum.3 . . . . . . . . . . . 12 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
13 fsum.4 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
14 fsum.5 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
1510, 11, 12, 13, 14fsumgcl 11095 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
1615ad2antrr 477 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
17 1zzd 9032 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ∈ ℤ)
1811nnzd 9123 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
1918ad2antrr 477 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑀 ∈ ℤ)
20 eluzelz 9284 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℤ)
2120ad2antlr 478 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ ℤ)
2217, 19, 213jca 1144 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ))
23 eluzle 9287 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 1 ≤ 𝑥)
2423ad2antlr 478 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ≤ 𝑥)
25 simpr 109 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥𝑀)
2624, 25jca 302 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ≤ 𝑥𝑥𝑀))
27 elfz2 9737 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥𝑥𝑀)))
2822, 26, 27sylanbrc 411 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ (1...𝑀))
299, 16, 28rspcdva 2766 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (𝐺𝑥) ∈ ℂ)
30 0cnd 7723 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ ¬ 𝑥𝑀) → 0 ∈ ℂ)
317nnzd 9123 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℤ)
3218adantr 272 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑀 ∈ ℤ)
33 zdcle 9078 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑥𝑀)
3431, 32, 33syl2anc 406 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ‘1)) → DECID 𝑥𝑀)
3529, 30, 34ifcldadc 3469 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ)
36 breq1 3900 . . . . . . . . . 10 (𝑛 = 𝑥 → (𝑛𝑀𝑥𝑀))
3736, 8ifbieq1d 3462 . . . . . . . . 9 (𝑛 = 𝑥 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑥𝑀, (𝐺𝑥), 0))
38 eqid 2115 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))
3937, 38fvmptg 5463 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
407, 35, 39syl2anc 406 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
4140, 35eqeltrd 2192 . . . . . 6 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) ∈ ℂ)
424, 41sylan2b 283 . . . . 5 ((𝜑𝑥 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) ∈ ℂ)
432, 3, 42serf 10187 . . . 4 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))):ℕ⟶ℂ)
4443, 11ffvelrnd 5522 . . 3 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) ∈ ℂ)
4544adantr 272 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) ∈ ℂ)
46 eleq1w 2176 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝑛𝐴𝑗𝐴))
47 csbeq1 2976 . . . . . . . . . . . . 13 (𝑛 = 𝑗𝑛 / 𝑘𝐵 = 𝑗 / 𝑘𝐵)
4846, 47ifbieq1d 3462 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
4948cbvmptv 3992 . . . . . . . . . . 11 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
5013ralrimiva 2480 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
51 nfcsb1v 3003 . . . . . . . . . . . . . 14 𝑘𝑗 / 𝑘𝐵
5251nfel1 2267 . . . . . . . . . . . . 13 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
53 csbeq1a 2981 . . . . . . . . . . . . . 14 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
5453eleq1d 2184 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
5552, 54rspc 2755 . . . . . . . . . . . 12 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
5650, 55mpan9 277 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
57 breq1 3900 . . . . . . . . . . . . 13 (𝑛 = 𝑖 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
58 fveq2 5387 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
5958csbeq1d 2979 . . . . . . . . . . . . . 14 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
60 csbco 2982 . . . . . . . . . . . . . 14 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
6159, 60syl6eqr 2166 . . . . . . . . . . . . 13 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
6257, 61ifbieq1d 3462 . . . . . . . . . . . 12 (𝑛 = 𝑖 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6362cbvmptv 3992 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑖 ∈ ℕ ↦ if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6449, 56, 63, 63summodc 11092 . . . . . . . . . 10 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
65 eleq1w 2176 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑗 → (𝑢𝐴𝑗𝐴))
6665dcbid 806 . . . . . . . . . . . . . . 15 (𝑢 = 𝑗 → (DECID 𝑢𝐴DECID 𝑗𝐴))
6766cbvralv 2629 . . . . . . . . . . . . . 14 (∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
68673anbi2i 1156 . . . . . . . . . . . . 13 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
6968rexbii 2417 . . . . . . . . . . . 12 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
70 1zzd 9032 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 1 ∈ ℤ)
71 nnz 9024 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
7270, 71fzfigd 10144 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → (1...𝑚) ∈ Fin)
73 fihasheqf1oi 10474 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
7472, 73sylan 279 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
75 nnnn0 8935 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7675adantr 272 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ0)
77 hashfz1 10469 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
7876, 77syl 14 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = 𝑚)
7974, 78eqtr3d 2150 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘𝐴) = 𝑚)
8079breq2d 3909 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛𝑚))
8180ifbid 3461 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
8281mpteq2dv 3987 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))
8382seqeq3d 10166 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))))
8483fveq1d 5389 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
8584eqeq2d 2127 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
8685pm5.32da 445 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
8786exbidv 1779 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
8887rexbiia 2425 . . . . . . . . . . . 12 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
8969, 88orbi12i 736 . . . . . . . . . . 11 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
9089mobii 2012 . . . . . . . . . 10 (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ↔ ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
9164, 90sylib 121 . . . . . . . . 9 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
9291adantr 272 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
93 simpr 109 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
94 f1of 5333 . . . . . . . . . . . . . 14 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
9512, 94syl 14 . . . . . . . . . . . . 13 (𝜑𝐹:(1...𝑀)⟶𝐴)
963, 18fzfigd 10144 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ∈ Fin)
97 fex 5613 . . . . . . . . . . . . 13 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin) → 𝐹 ∈ V)
9895, 96, 97syl2anc 406 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
9911, 2syl6eleq 2208 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ‘1))
10014ralrimiva 2480 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = 𝐶)
101 nfv 1491 . . . . . . . . . . . . . . . . . 18 𝑘(𝐺𝑛) = 𝐶
102 nfcsb1v 3003 . . . . . . . . . . . . . . . . . . 19 𝑛𝑘 / 𝑛𝐶
103102nfeq2 2268 . . . . . . . . . . . . . . . . . 18 𝑛(𝐺𝑘) = 𝑘 / 𝑛𝐶
104 fveq2 5387 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
105 csbeq1a 2981 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘𝐶 = 𝑘 / 𝑛𝐶)
106104, 105eqeq12d 2130 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝐺𝑛) = 𝐶 ↔ (𝐺𝑘) = 𝑘 / 𝑛𝐶))
107101, 103, 106cbvral 2625 . . . . . . . . . . . . . . . . 17 (∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = 𝐶 ↔ ∀𝑘 ∈ (1...𝑀)(𝐺𝑘) = 𝑘 / 𝑛𝐶)
108100, 107sylib 121 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ (1...𝑀)(𝐺𝑘) = 𝑘 / 𝑛𝐶)
109108r19.21bi 2495 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) = 𝑘 / 𝑛𝐶)
110 elfznn 9774 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
111110adantl 273 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ ℕ)
112 elfzle2 9748 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑀) → 𝑘𝑀)
113112adantl 273 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘𝑀)
114113iftrued 3449 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) = (𝐺𝑘))
115104eleq1d 2184 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑘) ∈ ℂ))
11615adantr 272 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
117 simpr 109 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ (1...𝑀))
118115, 116, 117rspcdva 2766 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) ∈ ℂ)
119114, 118eqeltrd 2192 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ)
120 breq1 3900 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛𝑀𝑘𝑀))
121120, 104ifbieq1d 3462 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑘𝑀, (𝐺𝑘), 0))
122121, 38fvmptg 5463 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
123111, 119, 122syl2anc 406 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
124123, 114eqtrd 2148 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = (𝐺𝑘))
125113iftrued 3449 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) = 𝑘 / 𝑛𝐶)
12695ffvelrnda 5521 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
12710adantl 273 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
128126, 127csbied 3014 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
12950adantr 272 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
130 nfcsb1v 3003 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐹𝑛) / 𝑘𝐵
131130nfel1 2267 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
132 csbeq1a 2981 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
133132eleq1d 2184 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
134131, 133rspc 2755 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
135126, 129, 134sylc 62 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
136128, 135eqeltrrd 2193 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
137136ralrimiva 2480 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
138 nfv 1491 . . . . . . . . . . . . . . . . . . . . 21 𝑘 𝐶 ∈ ℂ
139102nfel1 2267 . . . . . . . . . . . . . . . . . . . . 21 𝑛𝑘 / 𝑛𝐶 ∈ ℂ
140105eleq1d 2184 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝐶 ∈ ℂ ↔ 𝑘 / 𝑛𝐶 ∈ ℂ))
141138, 139, 140cbvral 2625 . . . . . . . . . . . . . . . . . . . 20 (∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ ↔ ∀𝑘 ∈ (1...𝑀)𝑘 / 𝑛𝐶 ∈ ℂ)
142137, 141sylib 121 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (1...𝑀)𝑘 / 𝑛𝐶 ∈ ℂ)
143142r19.21bi 2495 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 / 𝑛𝐶 ∈ ℂ)
144125, 143eqeltrd 2192 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ)
145 nfcv 2256 . . . . . . . . . . . . . . . . . 18 𝑛𝑘
146 nfv 1491 . . . . . . . . . . . . . . . . . . 19 𝑛 𝑘𝑀
147 nfcv 2256 . . . . . . . . . . . . . . . . . . 19 𝑛0
148146, 102, 147nfif 3468 . . . . . . . . . . . . . . . . . 18 𝑛if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0)
149120, 105ifbieq1d 3462 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, 𝐶, 0) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
150 eqid 2115 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))
151145, 148, 149, 150fvmptf 5479 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
152111, 144, 151syl2anc 406 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
153152, 125eqtrd 2148 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = 𝑘 / 𝑛𝐶)
154109, 124, 1533eqtr4d 2158 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘))
155137ad2antrr 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
156 nfcsb1v 3003 . . . . . . . . . . . . . . . . . . . 20 𝑛𝑥 / 𝑛𝐶
157156nfel1 2267 . . . . . . . . . . . . . . . . . . 19 𝑛𝑥 / 𝑛𝐶 ∈ ℂ
158 csbeq1a 2981 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑥𝐶 = 𝑥 / 𝑛𝐶)
159158eleq1d 2184 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑥 → (𝐶 ∈ ℂ ↔ 𝑥 / 𝑛𝐶 ∈ ℂ))
160157, 159rspc 2755 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (1...𝑀) → (∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ → 𝑥 / 𝑛𝐶 ∈ ℂ))
16128, 155, 160sylc 62 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 / 𝑛𝐶 ∈ ℂ)
162161, 30, 34ifcldadc 3469 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ)
163 nfcv 2256 . . . . . . . . . . . . . . . . 17 𝑛𝑥
164 nfv 1491 . . . . . . . . . . . . . . . . . 18 𝑛 𝑥𝑀
165164, 156, 147nfif 3468 . . . . . . . . . . . . . . . . 17 𝑛if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0)
16636, 158ifbieq1d 3462 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑥 → if(𝑛𝑀, 𝐶, 0) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
167163, 165, 166, 150fvmptf 5479 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
1687, 162, 167syl2anc 406 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
169168, 162eqeltrd 2192 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) ∈ ℂ)
170 addcl 7709 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
171170adantl 273 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
17299, 154, 41, 169, 171seq3fveq 10184 . . . . . . . . . . . . 13 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀))
17312, 172jca 302 . . . . . . . . . . . 12 (𝜑 → (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀)))
174 f1oeq1 5324 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
175 fveq1 5386 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
176175csbeq1d 2979 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
177 vex 2661 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 ∈ V
178 vex 2661 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛 ∈ V
179177, 178fvex 5407 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑛) ∈ V
180175, 179syl6eqelr 2207 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝐹𝑛) ∈ V)
18110adantl 273 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 = 𝐹𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
182180, 181csbied 3014 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝐹𝑛) / 𝑘𝐵 = 𝐶)
183176, 182eqtrd 2148 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = 𝐶)
184183ifeq1d 3457 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝐹 → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, 𝐶, 0))
185184mpteq2dv 3987 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))
186185seqeq3d 10166 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))))
187186fveq1d 5389 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀))
188187eqeq2d 2127 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → ((seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀) ↔ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀)))
189174, 188anbi12d 462 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀)) ↔ (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀))))
190189spcegv 2746 . . . . . . . . . . . 12 (𝐹 ∈ V → ((𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀)) → ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀))))
19198, 173, 190sylc 62 . . . . . . . . . . 11 (𝜑 → ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀)))
192 oveq2 5748 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
193 f1oeq2 5325 . . . . . . . . . . . . . . 15 ((1...𝑚) = (1...𝑀) → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
194192, 193syl 14 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
195 breq2 3901 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → (𝑛𝑚𝑛𝑀))
196195ifbid 3461 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑀 → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
197196mpteq2dv 3987 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑀 → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))
198197seqeq3d 10166 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑀 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))))
199 id 19 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑀𝑚 = 𝑀)
200198, 199fveq12d 5394 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀))
201200eqeq2d 2127 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → ((seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) ↔ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀)))
202194, 201anbi12d 462 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → ((𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ (𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀))))
203202exbidv 1779 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀))))
204203rspcev 2761 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀))) → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
20511, 191, 204syl2anc 406 . . . . . . . . . 10 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
206205olcd 706 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
207206adantr 272 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
208 breq2 3901 . . . . . . . . . . . 12 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)))
2092083anbi3d 1279 . . . . . . . . . . 11 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))))
210209rexbidv 2413 . . . . . . . . . 10 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))))
211 eqeq1 2122 . . . . . . . . . . . . 13 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) ↔ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
212211anbi2d 457 . . . . . . . . . . . 12 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
213212exbidv 1779 . . . . . . . . . . 11 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
214213rexbidv 2413 . . . . . . . . . 10 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
215210, 214orbi12d 765 . . . . . . . . 9 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))))
216215moi2 2836 . . . . . . . 8 ((((seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) ∈ ℂ ∧ ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) ∧ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))) → 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
21745, 92, 93, 207, 216syl22anc 1200 . . . . . . 7 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) → 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
218217ex 114 . . . . . 6 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) → 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)))
219206, 215syl5ibrcom 156 . . . . . 6 (𝜑 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))))
220218, 219impbid 128 . . . . 5 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)))
221220adantr 272 . . . 4 ((𝜑 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) ∈ ℂ) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀)))
222221iota5 5076 . . 3 ((𝜑 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) ∈ ℂ) → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
22344, 222mpdan 415 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
2241, 223syl5eq 2160 1 (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802  w3a 945   = wceq 1314  wex 1451  wcel 1463  ∃*wmo 1976  wral 2391  wrex 2392  Vcvv 2658  csb 2973  wss 3039  ifcif 3442   class class class wbr 3897  cmpt 3957  cio 5054  wf 5087  1-1-ontowf1o 5090  cfv 5091  (class class class)co 5740  Fincfn 6600  cc 7582  0cc0 7584  1c1 7585   + caddc 7587  cle 7765  cn 8677  0cn0 8928  cz 9005  cuz 9275  ...cfz 9730  seqcseq 10158  chash 10461  cli 10987  Σcsu 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063
This theorem is referenced by:  isumz  11098  fsumf1o  11099  fsumcl2lem  11107  fsumadd  11115  sumsnf  11118  fsummulc2  11157
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