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Theorem fisum 10742
 Description: The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 14-Sep-2022.) Use fsum3 10743 instead. (New usage is discouraged.)
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
fisum (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘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 fisum
Dummy variables 𝑓 𝑖 𝑗 𝑚 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isum 10707 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
2 fsum.2 . . . . 5 (𝜑𝑀 ∈ ℕ)
3 nnuz 9023 . . . . 5 ℕ = (ℤ‘1)
42, 3syl6eleq 2180 . . . 4 (𝜑𝑀 ∈ (ℤ‘1))
53eqimss2i 3079 . . . . . . . 8 (ℤ‘1) ⊆ ℕ
65sseli 3019 . . . . . . 7 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℕ)
76adantl 271 . . . . . 6 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℕ)
8 fveq2 5289 . . . . . . . . 9 (𝑛 = 𝑥 → (𝐺𝑛) = (𝐺𝑥))
98eleq1d 2156 . . . . . . . 8 (𝑛 = 𝑥 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑥) ∈ ℂ))
10 fsum.1 . . . . . . . . . 10 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
11 fsum.3 . . . . . . . . . 10 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
12 fsum.4 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
13 fsum.5 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
1410, 2, 11, 12, 13fsumgcl 10741 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
1514ad2antrr 472 . . . . . . . 8 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
16 1zzd 8747 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ∈ ℤ)
172nnzd 8837 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
1817ad2antrr 472 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑀 ∈ ℤ)
19 eluzelz 8997 . . . . . . . . . . 11 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℤ)
2019ad2antlr 473 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ ℤ)
2116, 18, 203jca 1123 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ))
22 eluzle 9000 . . . . . . . . . . 11 (𝑥 ∈ (ℤ‘1) → 1 ≤ 𝑥)
2322ad2antlr 473 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ≤ 𝑥)
24 simpr 108 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥𝑀)
2523, 24jca 300 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ≤ 𝑥𝑥𝑀))
26 elfz2 9400 . . . . . . . . 9 (𝑥 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥𝑥𝑀)))
2721, 25, 26sylanbrc 408 . . . . . . . 8 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ (1...𝑀))
289, 15, 27rspcdva 2727 . . . . . . 7 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (𝐺𝑥) ∈ ℂ)
29 0cnd 7460 . . . . . . 7 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ ¬ 𝑥𝑀) → 0 ∈ ℂ)
307nnzd 8837 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℤ)
3117adantr 270 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑀 ∈ ℤ)
32 zdcle 8793 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑥𝑀)
3330, 31, 32syl2anc 403 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ‘1)) → DECID 𝑥𝑀)
3428, 29, 33ifcldadc 3416 . . . . . 6 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ)
35 breq1 3840 . . . . . . . 8 (𝑛 = 𝑥 → (𝑛𝑀𝑥𝑀))
3635, 8ifbieq1d 3409 . . . . . . 7 (𝑛 = 𝑥 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑥𝑀, (𝐺𝑥), 0))
37 eqid 2088 . . . . . . 7 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))
3836, 37fvmptg 5364 . . . . . 6 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
397, 34, 38syl2anc 403 . . . . 5 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
4039, 34eqeltrd 2164 . . . 4 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) ∈ ℂ)
41 addcl 7446 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
4241adantl 271 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
434, 40, 42iseqcl 9846 . . 3 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) ∈ ℂ)
4443adantr 270 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) ∈ ℂ)
45 eleq1w 2148 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝑛𝐴𝑗𝐴))
46 csbeq1 2934 . . . . . . . . . . . . 13 (𝑛 = 𝑗𝑛 / 𝑘𝐵 = 𝑗 / 𝑘𝐵)
4745, 46ifbieq1d 3409 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
4847cbvmptv 3926 . . . . . . . . . . 11 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
4912ralrimiva 2446 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
50 nfcsb1v 2961 . . . . . . . . . . . . . 14 𝑘𝑗 / 𝑘𝐵
5150nfel1 2239 . . . . . . . . . . . . 13 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
52 csbeq1a 2939 . . . . . . . . . . . . . 14 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
5352eleq1d 2156 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
5451, 53rspc 2716 . . . . . . . . . . . 12 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
5549, 54mpan9 275 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
56 breq1 3840 . . . . . . . . . . . . 13 (𝑛 = 𝑖 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
57 fveq2 5289 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
5857csbeq1d 2937 . . . . . . . . . . . . . 14 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
59 csbco 2940 . . . . . . . . . . . . . 14 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
6058, 59syl6eqr 2138 . . . . . . . . . . . . 13 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
6156, 60ifbieq1d 3409 . . . . . . . . . . . 12 (𝑛 = 𝑖 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6261cbvmptv 3926 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑖 ∈ ℕ ↦ if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6348, 55, 62isummo 10737 . . . . . . . . . 10 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
64 eleq1w 2148 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑗 → (𝑢𝐴𝑗𝐴))
6564dcbid 786 . . . . . . . . . . . . . . 15 (𝑢 = 𝑗 → (DECID 𝑢𝐴DECID 𝑗𝐴))
6665cbvralv 2590 . . . . . . . . . . . . . 14 (∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
67663anbi2i 1135 . . . . . . . . . . . . 13 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
6867rexbii 2385 . . . . . . . . . . . 12 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
69 1zzd 8747 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 1 ∈ ℤ)
70 nnz 8739 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
7169, 70fzfigd 9803 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → (1...𝑚) ∈ Fin)
72 fihasheqf1oi 10161 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
7371, 72sylan 277 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
74 nnnn0 8650 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7574adantr 270 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ0)
76 hashfz1 10156 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
7775, 76syl 14 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = 𝑚)
7873, 77eqtr3d 2122 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘𝐴) = 𝑚)
7978breq2d 3849 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛𝑚))
8079ifbid 3408 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
8180mpteq2dv 3921 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))
82 iseqeq3 9825 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ))
8381, 82syl 14 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ))
8483fveq1d 5291 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
8584eqeq2d 2099 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
8685pm5.32da 440 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
8786exbidv 1753 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
8887rexbiia 2393 . . . . . . . . . . . 12 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
8968, 88orbi12i 716 . . . . . . . . . . 11 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
9089mobii 1985 . . . . . . . . . 10 (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
9163, 90sylib 120 . . . . . . . . 9 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
9291adantr 270 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
93 simpr 108 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
94 f1of 5237 . . . . . . . . . . . . . 14 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
9511, 94syl 14 . . . . . . . . . . . . 13 (𝜑𝐹:(1...𝑀)⟶𝐴)
96 1zzd 8747 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ ℤ)
9796, 17fzfigd 9803 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ∈ Fin)
98 fex 5506 . . . . . . . . . . . . 13 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin) → 𝐹 ∈ V)
9995, 97, 98syl2anc 403 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
10013ralrimiva 2446 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = 𝐶)
101 nfv 1466 . . . . . . . . . . . . . . . . . 18 𝑘(𝐺𝑛) = 𝐶
102 nfcsb1v 2961 . . . . . . . . . . . . . . . . . . 19 𝑛𝑘 / 𝑛𝐶
103102nfeq2 2240 . . . . . . . . . . . . . . . . . 18 𝑛(𝐺𝑘) = 𝑘 / 𝑛𝐶
104 fveq2 5289 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
105 csbeq1a 2939 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘𝐶 = 𝑘 / 𝑛𝐶)
106104, 105eqeq12d 2102 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝐺𝑛) = 𝐶 ↔ (𝐺𝑘) = 𝑘 / 𝑛𝐶))
107101, 103, 106cbvral 2586 . . . . . . . . . . . . . . . . 17 (∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = 𝐶 ↔ ∀𝑘 ∈ (1...𝑀)(𝐺𝑘) = 𝑘 / 𝑛𝐶)
108100, 107sylib 120 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ (1...𝑀)(𝐺𝑘) = 𝑘 / 𝑛𝐶)
109108r19.21bi 2461 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) = 𝑘 / 𝑛𝐶)
110 elfznn 9437 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
111110adantl 271 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ ℕ)
112 elfzle2 9411 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑀) → 𝑘𝑀)
113112adantl 271 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘𝑀)
114113iftrued 3396 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) = (𝐺𝑘))
115104eleq1d 2156 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑘) ∈ ℂ))
11614adantr 270 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
117 simpr 108 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ (1...𝑀))
118115, 116, 117rspcdva 2727 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) ∈ ℂ)
119114, 118eqeltrd 2164 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ)
120 breq1 3840 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛𝑀𝑘𝑀))
121120, 104ifbieq1d 3409 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑘𝑀, (𝐺𝑘), 0))
122121, 37fvmptg 5364 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
123111, 119, 122syl2anc 403 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
124123, 114eqtrd 2120 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = (𝐺𝑘))
125113iftrued 3396 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) = 𝑘 / 𝑛𝐶)
12695ffvelrnda 5418 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
12710adantl 271 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
128126, 127csbied 2972 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
12949adantr 270 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
130 nfcsb1v 2961 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐹𝑛) / 𝑘𝐵
131130nfel1 2239 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
132 csbeq1a 2939 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
133132eleq1d 2156 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
134131, 133rspc 2716 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
135126, 129, 134sylc 61 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
136128, 135eqeltrrd 2165 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
137136ralrimiva 2446 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
138 nfv 1466 . . . . . . . . . . . . . . . . . . . . 21 𝑘 𝐶 ∈ ℂ
139102nfel1 2239 . . . . . . . . . . . . . . . . . . . . 21 𝑛𝑘 / 𝑛𝐶 ∈ ℂ
140105eleq1d 2156 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝐶 ∈ ℂ ↔ 𝑘 / 𝑛𝐶 ∈ ℂ))
141138, 139, 140cbvral 2586 . . . . . . . . . . . . . . . . . . . 20 (∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ ↔ ∀𝑘 ∈ (1...𝑀)𝑘 / 𝑛𝐶 ∈ ℂ)
142137, 141sylib 120 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (1...𝑀)𝑘 / 𝑛𝐶 ∈ ℂ)
143142r19.21bi 2461 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 / 𝑛𝐶 ∈ ℂ)
144125, 143eqeltrd 2164 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ)
145 nfcv 2228 . . . . . . . . . . . . . . . . . 18 𝑛𝑘
146 nfv 1466 . . . . . . . . . . . . . . . . . . 19 𝑛 𝑘𝑀
147 nfcv 2228 . . . . . . . . . . . . . . . . . . 19 𝑛0
148146, 102, 147nfif 3415 . . . . . . . . . . . . . . . . . 18 𝑛if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0)
149120, 105ifbieq1d 3409 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, 𝐶, 0) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
150 eqid 2088 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))
151145, 148, 149, 150fvmptf 5379 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
152111, 144, 151syl2anc 403 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
153152, 125eqtrd 2120 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = 𝑘 / 𝑛𝐶)
154109, 124, 1533eqtr4d 2130 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘))
155137ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
156 nfcsb1v 2961 . . . . . . . . . . . . . . . . . . . 20 𝑛𝑥 / 𝑛𝐶
157156nfel1 2239 . . . . . . . . . . . . . . . . . . 19 𝑛𝑥 / 𝑛𝐶 ∈ ℂ
158 csbeq1a 2939 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑥𝐶 = 𝑥 / 𝑛𝐶)
159158eleq1d 2156 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑥 → (𝐶 ∈ ℂ ↔ 𝑥 / 𝑛𝐶 ∈ ℂ))
160157, 159rspc 2716 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (1...𝑀) → (∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ → 𝑥 / 𝑛𝐶 ∈ ℂ))
16127, 155, 160sylc 61 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 / 𝑛𝐶 ∈ ℂ)
162161, 29, 33ifcldadc 3416 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ)
163 nfcv 2228 . . . . . . . . . . . . . . . . 17 𝑛𝑥
164 nfv 1466 . . . . . . . . . . . . . . . . . 18 𝑛 𝑥𝑀
165164, 156, 147nfif 3415 . . . . . . . . . . . . . . . . 17 𝑛if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0)
16635, 158ifbieq1d 3409 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑥 → if(𝑛𝑀, 𝐶, 0) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
167163, 165, 166, 150fvmptf 5379 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
1687, 162, 167syl2anc 403 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
169168, 162eqeltrd 2164 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) ∈ ℂ)
1704, 154, 40, 169, 42iseqfveq 9859 . . . . . . . . . . . . 13 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ)‘𝑀))
17111, 170jca 300 . . . . . . . . . . . 12 (𝜑 → (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ)‘𝑀)))
172 f1oeq1 5228 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
173 fveq1 5288 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
174173csbeq1d 2937 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
175 vex 2622 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 ∈ V
176 vex 2622 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛 ∈ V
177175, 176fvex 5309 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑛) ∈ V
178173, 177syl6eqelr 2179 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝐹𝑛) ∈ V)
17910adantl 271 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 = 𝐹𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
180178, 179csbied 2972 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝐹𝑛) / 𝑘𝐵 = 𝐶)
181174, 180eqtrd 2120 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = 𝐶)
182181ifeq1d 3404 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝐹 → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, 𝐶, 0))
183182mpteq2dv 3921 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))
184 iseqeq3 9825 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ))
185183, 184syl 14 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ))
186185fveq1d 5291 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ)‘𝑀))
187186eqeq2d 2099 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → ((seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀) ↔ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ)‘𝑀)))
188172, 187anbi12d 457 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀)) ↔ (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ)‘𝑀))))
189188spcegv 2707 . . . . . . . . . . . 12 (𝐹 ∈ V → ((𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)), ℂ)‘𝑀)) → ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀))))
19099, 171, 189sylc 61 . . . . . . . . . . 11 (𝜑 → ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀)))
191 oveq2 5642 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
192 f1oeq2 5229 . . . . . . . . . . . . . . 15 ((1...𝑚) = (1...𝑀) → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
193191, 192syl 14 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
194 breq2 3841 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → (𝑛𝑚𝑛𝑀))
195194ifbid 3408 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑀 → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
196195mpteq2dv 3921 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑀 → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))
197 iseqeq3 9825 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ))
198196, 197syl 14 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑀 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ))
199 id 19 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑀𝑚 = 𝑀)
200198, 199fveq12d 5296 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀))
201200eqeq2d 2099 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → ((seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) ↔ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀)))
202193, 201anbi12d 457 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → ((𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ (𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀))))
203202exbidv 1753 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀))))
204203rspcev 2722 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑀))) → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
2052, 190, 204syl2anc 403 . . . . . . . . . 10 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
206205olcd 688 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
207206adantr 270 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
208 breq2 3841 . . . . . . . . . . . 12 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)))
2092083anbi3d 1254 . . . . . . . . . . 11 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀))))
210209rexbidv 2381 . . . . . . . . . 10 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀))))
211 eqeq1 2094 . . . . . . . . . . . . 13 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) ↔ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
212211anbi2d 452 . . . . . . . . . . . 12 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
213212exbidv 1753 . . . . . . . . . . 11 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
214213rexbidv 2381 . . . . . . . . . 10 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
215210, 214orbi12d 742 . . . . . . . . 9 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))))
216215moi2 2794 . . . . . . . 8 ((((seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) ∈ ℂ ∧ ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) ∧ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))) → 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀))
21744, 92, 93, 207, 216syl22anc 1175 . . . . . . 7 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) → 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀))
218217ex 113 . . . . . 6 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) → 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)))
219206, 215syl5ibrcom 155 . . . . . 6 (𝜑 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))))
220218, 219impbid 127 . . . . 5 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)))
221220adantr 270 . . . 4 ((𝜑 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) ∈ ℂ) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀)))
222221iota5 4987 . . 3 ((𝜑 ∧ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀) ∈ ℂ) → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀))
22343, 222mpdan 412 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀))
2241, 223syl5eq 2132 1 (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)), ℂ)‘𝑀))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 664  DECID wdc 780   ∧ w3a 924   = wceq 1289  ∃wex 1426   ∈ wcel 1438  ∃*wmo 1949  ∀wral 2359  ∃wrex 2360  Vcvv 2619  ⦋csb 2931   ⊆ wss 2997  ifcif 3389   class class class wbr 3837   ↦ cmpt 3891  ℩cio 4965  ⟶wf 4998  –1-1-onto→wf1o 5001  ‘cfv 5002  (class class class)co 5634  Fincfn 6437  ℂcc 7327  0cc0 7329  1c1 7330   + caddc 7332   ≤ cle 7502  ℕcn 8394  ℕ0cn0 8643  ℤcz 8720  ℤ≥cuz 8988  ...cfz 9393  seqcseq4 9816  ♯chash 10148   ⇝ cli 10630  Σcsu 10706 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444 This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-frec 6138  df-1o 6163  df-oadd 6167  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-iseq 9818  df-seq3 9819  df-exp 9920  df-ihash 10149  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-clim 10631  df-isum 10707 This theorem is referenced by:  fsum3  10743  isumz  10745  fsumf1o  10746  fsumadd  10763  sumsnf  10766
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