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Theorem fsum3 11261
 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 11228 . 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 9453 . . . . 5 ℕ = (ℤ‘1)
3 1zzd 9173 . . . . 5 (𝜑 → 1 ∈ ℤ)
4 elnnuz 9454 . . . . . 6 (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ‘1))
52eqimss2i 3181 . . . . . . . . . 10 (ℤ‘1) ⊆ ℕ
65sseli 3120 . . . . . . . . 9 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℕ)
76adantl 275 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℕ)
8 fveq2 5461 . . . . . . . . . . 11 (𝑛 = 𝑥 → (𝐺𝑛) = (𝐺𝑥))
98eleq1d 2223 . . . . . . . . . 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 11260 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
1615ad2antrr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
17 1zzd 9173 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ∈ ℤ)
1811nnzd 9264 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
1918ad2antrr 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑀 ∈ ℤ)
20 eluzelz 9427 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℤ)
2120ad2antlr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ ℤ)
2217, 19, 213jca 1162 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ))
23 eluzle 9430 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 1 ≤ 𝑥)
2423ad2antlr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ≤ 𝑥)
25 simpr 109 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥𝑀)
2624, 25jca 304 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ≤ 𝑥𝑥𝑀))
27 elfz2 9897 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥𝑥𝑀)))
2822, 26, 27sylanbrc 414 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ (1...𝑀))
299, 16, 28rspcdva 2818 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (𝐺𝑥) ∈ ℂ)
30 0cnd 7850 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ ¬ 𝑥𝑀) → 0 ∈ ℂ)
317nnzd 9264 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℤ)
3218adantr 274 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑀 ∈ ℤ)
33 zdcle 9219 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑥𝑀)
3431, 32, 33syl2anc 409 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ‘1)) → DECID 𝑥𝑀)
3529, 30, 34ifcldadc 3530 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ)
36 breq1 3964 . . . . . . . . . 10 (𝑛 = 𝑥 → (𝑛𝑀𝑥𝑀))
3736, 8ifbieq1d 3523 . . . . . . . . 9 (𝑛 = 𝑥 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑥𝑀, (𝐺𝑥), 0))
38 eqid 2154 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))
3937, 38fvmptg 5537 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
407, 35, 39syl2anc 409 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
4140, 35eqeltrd 2231 . . . . . 6 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) ∈ ℂ)
424, 41sylan2b 285 . . . . 5 ((𝜑𝑥 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) ∈ ℂ)
432, 3, 42serf 10351 . . . 4 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))):ℕ⟶ℂ)
4443, 11ffvelrnd 5596 . . 3 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) ∈ ℂ)
4544adantr 274 . . . . . . . 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 2215 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝑛𝐴𝑗𝐴))
47 csbeq1 3030 . . . . . . . . . . . . 13 (𝑛 = 𝑗𝑛 / 𝑘𝐵 = 𝑗 / 𝑘𝐵)
4846, 47ifbieq1d 3523 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
4948cbvmptv 4056 . . . . . . . . . . 11 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
5013ralrimiva 2527 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
51 nfcsb1v 3060 . . . . . . . . . . . . . 14 𝑘𝑗 / 𝑘𝐵
5251nfel1 2307 . . . . . . . . . . . . 13 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
53 csbeq1a 3036 . . . . . . . . . . . . . 14 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
5453eleq1d 2223 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
5552, 54rspc 2807 . . . . . . . . . . . 12 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
5650, 55mpan9 279 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
57 breq1 3964 . . . . . . . . . . . . 13 (𝑛 = 𝑖 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
58 fveq2 5461 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
5958csbeq1d 3034 . . . . . . . . . . . . . 14 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
60 csbco 3037 . . . . . . . . . . . . . 14 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
6159, 60eqtr4di 2205 . . . . . . . . . . . . 13 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
6257, 61ifbieq1d 3523 . . . . . . . . . . . 12 (𝑛 = 𝑖 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6362cbvmptv 4056 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑖 ∈ ℕ ↦ if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6449, 56, 63, 63summodc 11257 . . . . . . . . . 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 2215 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑗 → (𝑢𝐴𝑗𝐴))
6665dcbid 824 . . . . . . . . . . . . . . 15 (𝑢 = 𝑗 → (DECID 𝑢𝐴DECID 𝑗𝐴))
6766cbvralv 2677 . . . . . . . . . . . . . 14 (∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
68673anbi2i 1174 . . . . . . . . . . . . 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 2461 . . . . . . . . . . . 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 9173 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 1 ∈ ℤ)
71 nnz 9165 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
7270, 71fzfigd 10308 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → (1...𝑚) ∈ Fin)
73 fihasheqf1oi 10639 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
7472, 73sylan 281 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
75 nnnn0 9076 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7675adantr 274 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ0)
77 hashfz1 10634 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
7876, 77syl 14 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = 𝑚)
7974, 78eqtr3d 2189 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘𝐴) = 𝑚)
8079breq2d 3973 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛𝑚))
8180ifbid 3522 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
8281mpteq2dv 4051 . . . . . . . . . . . . . . . . . 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 10330 . . . . . . . . . . . . . . . . 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 5463 . . . . . . . . . . . . . . . 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 2166 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . 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 1802 . . . . . . . . . . . . 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 2469 . . . . . . . . . . . 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 754 . . . . . . . . . . 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 2040 . . . . . . . . . 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 274 . . . . . . . 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 5407 . . . . . . . . . . . . . 14 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
9512, 94syl 14 . . . . . . . . . . . . 13 (𝜑𝐹:(1...𝑀)⟶𝐴)
963, 18fzfigd 10308 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ∈ Fin)
97 fex 5687 . . . . . . . . . . . . 13 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin) → 𝐹 ∈ V)
9895, 96, 97syl2anc 409 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
9911, 2eleqtrdi 2247 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ‘1))
10014ralrimiva 2527 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = 𝐶)
101 nfv 1505 . . . . . . . . . . . . . . . . . 18 𝑘(𝐺𝑛) = 𝐶
102 nfcsb1v 3060 . . . . . . . . . . . . . . . . . . 19 𝑛𝑘 / 𝑛𝐶
103102nfeq2 2308 . . . . . . . . . . . . . . . . . 18 𝑛(𝐺𝑘) = 𝑘 / 𝑛𝐶
104 fveq2 5461 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
105 csbeq1a 3036 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘𝐶 = 𝑘 / 𝑛𝐶)
106104, 105eqeq12d 2169 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝐺𝑛) = 𝐶 ↔ (𝐺𝑘) = 𝑘 / 𝑛𝐶))
107101, 103, 106cbvral 2673 . . . . . . . . . . . . . . . . 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 2542 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) = 𝑘 / 𝑛𝐶)
110 elfznn 9934 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
111110adantl 275 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ ℕ)
112 elfzle2 9908 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑀) → 𝑘𝑀)
113112adantl 275 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘𝑀)
114113iftrued 3508 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) = (𝐺𝑘))
115104eleq1d 2223 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑘) ∈ ℂ))
11615adantr 274 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
117 simpr 109 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ (1...𝑀))
118115, 116, 117rspcdva 2818 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) ∈ ℂ)
119114, 118eqeltrd 2231 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ)
120 breq1 3964 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛𝑀𝑘𝑀))
121120, 104ifbieq1d 3523 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑘𝑀, (𝐺𝑘), 0))
122121, 38fvmptg 5537 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
123111, 119, 122syl2anc 409 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
124123, 114eqtrd 2187 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = (𝐺𝑘))
125113iftrued 3508 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) = 𝑘 / 𝑛𝐶)
12695ffvelrnda 5595 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
12710adantl 275 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
128126, 127csbied 3073 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
12950adantr 274 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
130 nfcsb1v 3060 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐹𝑛) / 𝑘𝐵
131130nfel1 2307 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
132 csbeq1a 3036 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
133132eleq1d 2223 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
134131, 133rspc 2807 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
135126, 129, 134sylc 62 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
136128, 135eqeltrrd 2232 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
137136ralrimiva 2527 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
138 nfv 1505 . . . . . . . . . . . . . . . . . . . . 21 𝑘 𝐶 ∈ ℂ
139102nfel1 2307 . . . . . . . . . . . . . . . . . . . . 21 𝑛𝑘 / 𝑛𝐶 ∈ ℂ
140105eleq1d 2223 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝐶 ∈ ℂ ↔ 𝑘 / 𝑛𝐶 ∈ ℂ))
141138, 139, 140cbvral 2673 . . . . . . . . . . . . . . . . . . . 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 2542 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 / 𝑛𝐶 ∈ ℂ)
144125, 143eqeltrd 2231 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ)
145 nfcv 2296 . . . . . . . . . . . . . . . . . 18 𝑛𝑘
146 nfv 1505 . . . . . . . . . . . . . . . . . . 19 𝑛 𝑘𝑀
147 nfcv 2296 . . . . . . . . . . . . . . . . . . 19 𝑛0
148146, 102, 147nfif 3529 . . . . . . . . . . . . . . . . . 18 𝑛if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0)
149120, 105ifbieq1d 3523 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, 𝐶, 0) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
150 eqid 2154 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))
151145, 148, 149, 150fvmptf 5553 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
152111, 144, 151syl2anc 409 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
153152, 125eqtrd 2187 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = 𝑘 / 𝑛𝐶)
154109, 124, 1533eqtr4d 2197 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘))
155137ad2antrr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
156 nfcsb1v 3060 . . . . . . . . . . . . . . . . . . . 20 𝑛𝑥 / 𝑛𝐶
157156nfel1 2307 . . . . . . . . . . . . . . . . . . 19 𝑛𝑥 / 𝑛𝐶 ∈ ℂ
158 csbeq1a 3036 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑥𝐶 = 𝑥 / 𝑛𝐶)
159158eleq1d 2223 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑥 → (𝐶 ∈ ℂ ↔ 𝑥 / 𝑛𝐶 ∈ ℂ))
160157, 159rspc 2807 . . . . . . . . . . . . . . . . . 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 3530 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ)
163 nfcv 2296 . . . . . . . . . . . . . . . . 17 𝑛𝑥
164 nfv 1505 . . . . . . . . . . . . . . . . . 18 𝑛 𝑥𝑀
165164, 156, 147nfif 3529 . . . . . . . . . . . . . . . . 17 𝑛if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0)
16636, 158ifbieq1d 3523 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑥 → if(𝑛𝑀, 𝐶, 0) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
167163, 165, 166, 150fvmptf 5553 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
1687, 162, 167syl2anc 409 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
169168, 162eqeltrd 2231 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) ∈ ℂ)
170 addcl 7836 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
171170adantl 275 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
17299, 154, 41, 169, 171seq3fveq 10348 . . . . . . . . . . . . 13 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀))
17312, 172jca 304 . . . . . . . . . . . 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 5396 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
175 fveq1 5460 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
176175csbeq1d 3034 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
177 vex 2712 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 ∈ V
178 vex 2712 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛 ∈ V
179177, 178fvex 5481 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑛) ∈ V
180175, 179eqeltrrdi 2246 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝐹𝑛) ∈ V)
18110adantl 275 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 = 𝐹𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
182180, 181csbied 3073 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝐹𝑛) / 𝑘𝐵 = 𝐶)
183176, 182eqtrd 2187 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = 𝐶)
184183ifeq1d 3518 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝐹 → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, 𝐶, 0))
185184mpteq2dv 4051 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))
186185seqeq3d 10330 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))))
187186fveq1d 5463 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀))
188187eqeq2d 2166 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 2797 . . . . . . . . . . . 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 5822 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
193 f1oeq2 5397 . . . . . . . . . . . . . . 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 3965 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → (𝑛𝑚𝑛𝑀))
196195ifbid 3522 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑀 → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
197196mpteq2dv 4051 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑀 → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))
198197seqeq3d 10330 . . . . . . . . . . . . . . . 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 5468 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀))
201200eqeq2d 2166 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 1802 . . . . . . . . . . . 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 2813 . . . . . . . . . . 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 409 . . . . . . . . . 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 724 . . . . . . . . 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 274 . . . . . . . 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 3965 . . . . . . . . . . . 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 1297 . . . . . . . . . . 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 2455 . . . . . . . . . 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 2161 . . . . . . . . . . . . 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 460 . . . . . . . . . . . 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 1802 . . . . . . . . . . 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 2455 . . . . . . . . . 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 783 . . . . . . . . 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 2889 . . . . . . . 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 1218 . . . . . . 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 274 . . . 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 5148 . . 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 418 . 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 2199 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 698  DECID wdc 820   ∧ w3a 963   = wceq 1332  ∃wex 1469  ∃*wmo 2004   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  Vcvv 2709  ⦋csb 3027   ⊆ wss 3098  ifcif 3501   class class class wbr 3961   ↦ cmpt 4021  ℩cio 5126  ⟶wf 5159  –1-1-onto→wf1o 5162  ‘cfv 5163  (class class class)co 5814  Fincfn 6674  ℂcc 7709  0cc0 7711  1c1 7712   + caddc 7714   ≤ cle 7892  ℕcn 8812  ℕ0cn0 9069  ℤcz 9146  ℤ≥cuz 9418  ...cfz 9890  seqcseq 10322  ♯chash 10626   ⇝ cli 11152  Σcsu 11227 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-irdg 6307  df-frec 6328  df-1o 6353  df-oadd 6357  df-er 6469  df-en 6675  df-dom 6676  df-fin 6677  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-fz 9891  df-fzo 10020  df-seqfrec 10323  df-exp 10397  df-ihash 10627  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-clim 11153  df-sumdc 11228 This theorem is referenced by:  isumz  11263  fsumf1o  11264  fsumcl2lem  11272  fsumadd  11280  sumsnf  11283  fsummulc2  11322
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