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Theorem fsum3 11379
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 11346 . 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 9552 . . . . 5 ℕ = (ℤ‘1)
3 1zzd 9269 . . . . 5 (𝜑 → 1 ∈ ℤ)
4 elnnuz 9553 . . . . . 6 (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ‘1))
52eqimss2i 3212 . . . . . . . . . 10 (ℤ‘1) ⊆ ℕ
65sseli 3151 . . . . . . . . 9 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℕ)
76adantl 277 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℕ)
8 fveq2 5511 . . . . . . . . . . 11 (𝑛 = 𝑥 → (𝐺𝑛) = (𝐺𝑥))
98eleq1d 2246 . . . . . . . . . 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 11378 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
1615ad2antrr 488 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
17 1zzd 9269 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ∈ ℤ)
1811nnzd 9363 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
1918ad2antrr 488 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑀 ∈ ℤ)
20 eluzelz 9526 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℤ)
2120ad2antlr 489 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ ℤ)
2217, 19, 213jca 1177 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ))
23 eluzle 9529 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 1 ≤ 𝑥)
2423ad2antlr 489 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 1 ≤ 𝑥)
25 simpr 110 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥𝑀)
2624, 25jca 306 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (1 ≤ 𝑥𝑥𝑀))
27 elfz2 10002 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥𝑥𝑀)))
2822, 26, 27sylanbrc 417 . . . . . . . . . 10 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → 𝑥 ∈ (1...𝑀))
299, 16, 28rspcdva 2846 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → (𝐺𝑥) ∈ ℂ)
30 0cnd 7941 . . . . . . . . 9 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ ¬ 𝑥𝑀) → 0 ∈ ℂ)
317nnzd 9363 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℤ)
3218adantr 276 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘1)) → 𝑀 ∈ ℤ)
33 zdcle 9318 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑥𝑀)
3431, 32, 33syl2anc 411 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ‘1)) → DECID 𝑥𝑀)
3529, 30, 34ifcldadc 3563 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ)
36 breq1 4003 . . . . . . . . . 10 (𝑛 = 𝑥 → (𝑛𝑀𝑥𝑀))
3736, 8ifbieq1d 3556 . . . . . . . . 9 (𝑛 = 𝑥 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑥𝑀, (𝐺𝑥), 0))
38 eqid 2177 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))
3937, 38fvmptg 5588 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, (𝐺𝑥), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
407, 35, 39syl2anc 411 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) = if(𝑥𝑀, (𝐺𝑥), 0))
4140, 35eqeltrd 2254 . . . . . 6 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) ∈ ℂ)
424, 41sylan2b 287 . . . . 5 ((𝜑𝑥 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑥) ∈ ℂ)
432, 3, 42serf 10460 . . . 4 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))):ℕ⟶ℂ)
4443, 11ffvelcdmd 5648 . . 3 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) ∈ ℂ)
4544adantr 276 . . . . . . . 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 2238 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝑛𝐴𝑗𝐴))
47 csbeq1 3060 . . . . . . . . . . . . 13 (𝑛 = 𝑗𝑛 / 𝑘𝐵 = 𝑗 / 𝑘𝐵)
4846, 47ifbieq1d 3556 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
4948cbvmptv 4096 . . . . . . . . . . 11 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
5013ralrimiva 2550 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
51 nfcsb1v 3090 . . . . . . . . . . . . . 14 𝑘𝑗 / 𝑘𝐵
5251nfel1 2330 . . . . . . . . . . . . 13 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
53 csbeq1a 3066 . . . . . . . . . . . . . 14 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
5453eleq1d 2246 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
5552, 54rspc 2835 . . . . . . . . . . . 12 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
5650, 55mpan9 281 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
57 breq1 4003 . . . . . . . . . . . . 13 (𝑛 = 𝑖 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
58 fveq2 5511 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
5958csbeq1d 3064 . . . . . . . . . . . . . 14 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
60 csbco 3067 . . . . . . . . . . . . . 14 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
6159, 60eqtr4di 2228 . . . . . . . . . . . . 13 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
6257, 61ifbieq1d 3556 . . . . . . . . . . . 12 (𝑛 = 𝑖 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6362cbvmptv 4096 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑖 ∈ ℕ ↦ if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 0))
6449, 56, 63, 63summodc 11375 . . . . . . . . . 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 2238 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑗 → (𝑢𝐴𝑗𝐴))
6665dcbid 838 . . . . . . . . . . . . . . 15 (𝑢 = 𝑗 → (DECID 𝑢𝐴DECID 𝑗𝐴))
6766cbvralv 2703 . . . . . . . . . . . . . 14 (∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
68673anbi2i 1191 . . . . . . . . . . . . 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 2484 . . . . . . . . . . . 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 9269 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 1 ∈ ℤ)
71 nnz 9261 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
7270, 71fzfigd 10417 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → (1...𝑚) ∈ Fin)
73 fihasheqf1oi 10751 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
7472, 73sylan 283 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
75 nnnn0 9172 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7675adantr 276 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ0)
77 hashfz1 10747 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
7876, 77syl 14 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = 𝑚)
7974, 78eqtr3d 2212 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘𝐴) = 𝑚)
8079breq2d 4012 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛𝑚))
8180ifbid 3555 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
8281mpteq2dv 4091 . . . . . . . . . . . . . . . . . 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 10439 . . . . . . . . . . . . . . . . 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 5513 . . . . . . . . . . . . . . . 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 2189 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 1825 . . . . . . . . . . . . 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 2492 . . . . . . . . . . . 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 764 . . . . . . . . . . 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 2063 . . . . . . . . . 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 122 . . . . . . . . 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 276 . . . . . . . 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 110 . . . . . . . 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 5457 . . . . . . . . . . . . . 14 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
9512, 94syl 14 . . . . . . . . . . . . 13 (𝜑𝐹:(1...𝑀)⟶𝐴)
963, 18fzfigd 10417 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ∈ Fin)
97 fex 5741 . . . . . . . . . . . . 13 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin) → 𝐹 ∈ V)
9895, 96, 97syl2anc 411 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
9911, 2eleqtrdi 2270 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ‘1))
10014ralrimiva 2550 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = 𝐶)
101 nfv 1528 . . . . . . . . . . . . . . . . . 18 𝑘(𝐺𝑛) = 𝐶
102 nfcsb1v 3090 . . . . . . . . . . . . . . . . . . 19 𝑛𝑘 / 𝑛𝐶
103102nfeq2 2331 . . . . . . . . . . . . . . . . . 18 𝑛(𝐺𝑘) = 𝑘 / 𝑛𝐶
104 fveq2 5511 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
105 csbeq1a 3066 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘𝐶 = 𝑘 / 𝑛𝐶)
106104, 105eqeq12d 2192 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝐺𝑛) = 𝐶 ↔ (𝐺𝑘) = 𝑘 / 𝑛𝐶))
107101, 103, 106cbvral 2699 . . . . . . . . . . . . . . . . 17 (∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = 𝐶 ↔ ∀𝑘 ∈ (1...𝑀)(𝐺𝑘) = 𝑘 / 𝑛𝐶)
108100, 107sylib 122 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ (1...𝑀)(𝐺𝑘) = 𝑘 / 𝑛𝐶)
109108r19.21bi 2565 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) = 𝑘 / 𝑛𝐶)
110 elfznn 10040 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
111110adantl 277 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ ℕ)
112 elfzle2 10014 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑀) → 𝑘𝑀)
113112adantl 277 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘𝑀)
114113iftrued 3541 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) = (𝐺𝑘))
115104eleq1d 2246 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑘) ∈ ℂ))
11615adantr 276 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
117 simpr 110 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 ∈ (1...𝑀))
118115, 116, 117rspcdva 2846 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) ∈ ℂ)
119114, 118eqeltrd 2254 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ)
120 breq1 4003 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛𝑀𝑘𝑀))
121120, 104ifbieq1d 3556 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, (𝐺𝑛), 0) = if(𝑘𝑀, (𝐺𝑘), 0))
122121, 38fvmptg 5588 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, (𝐺𝑘), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
123111, 119, 122syl2anc 411 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = if(𝑘𝑀, (𝐺𝑘), 0))
124123, 114eqtrd 2210 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = (𝐺𝑘))
125113iftrued 3541 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) = 𝑘 / 𝑛𝐶)
12695ffvelcdmda 5647 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
12710adantl 277 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
128126, 127csbied 3103 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
12950adantr 276 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
130 nfcsb1v 3090 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐹𝑛) / 𝑘𝐵
131130nfel1 2330 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
132 csbeq1a 3066 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
133132eleq1d 2246 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
134131, 133rspc 2835 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
135126, 129, 134sylc 62 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
136128, 135eqeltrrd 2255 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
137136ralrimiva 2550 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
138 nfv 1528 . . . . . . . . . . . . . . . . . . . . 21 𝑘 𝐶 ∈ ℂ
139102nfel1 2330 . . . . . . . . . . . . . . . . . . . . 21 𝑛𝑘 / 𝑛𝐶 ∈ ℂ
140105eleq1d 2246 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝐶 ∈ ℂ ↔ 𝑘 / 𝑛𝐶 ∈ ℂ))
141138, 139, 140cbvral 2699 . . . . . . . . . . . . . . . . . . . 20 (∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ ↔ ∀𝑘 ∈ (1...𝑀)𝑘 / 𝑛𝐶 ∈ ℂ)
142137, 141sylib 122 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (1...𝑀)𝑘 / 𝑛𝐶 ∈ ℂ)
143142r19.21bi 2565 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝑀)) → 𝑘 / 𝑛𝐶 ∈ ℂ)
144125, 143eqeltrd 2254 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...𝑀)) → if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ)
145 nfcv 2319 . . . . . . . . . . . . . . . . . 18 𝑛𝑘
146 nfv 1528 . . . . . . . . . . . . . . . . . . 19 𝑛 𝑘𝑀
147 nfcv 2319 . . . . . . . . . . . . . . . . . . 19 𝑛0
148146, 102, 147nfif 3562 . . . . . . . . . . . . . . . . . 18 𝑛if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0)
149120, 105ifbieq1d 3556 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → if(𝑛𝑀, 𝐶, 0) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
150 eqid 2177 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))
151145, 148, 149, 150fvmptf 5604 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
152111, 144, 151syl2anc 411 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = if(𝑘𝑀, 𝑘 / 𝑛𝐶, 0))
153152, 125eqtrd 2210 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘) = 𝑘 / 𝑛𝐶)
154109, 124, 1533eqtr4d 2220 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0))‘𝑘) = ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑘))
155137ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (ℤ‘1)) ∧ 𝑥𝑀) → ∀𝑛 ∈ (1...𝑀)𝐶 ∈ ℂ)
156 nfcsb1v 3090 . . . . . . . . . . . . . . . . . . . 20 𝑛𝑥 / 𝑛𝐶
157156nfel1 2330 . . . . . . . . . . . . . . . . . . 19 𝑛𝑥 / 𝑛𝐶 ∈ ℂ
158 csbeq1a 3066 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑥𝐶 = 𝑥 / 𝑛𝐶)
159158eleq1d 2246 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑥 → (𝐶 ∈ ℂ ↔ 𝑥 / 𝑛𝐶 ∈ ℂ))
160157, 159rspc 2835 . . . . . . . . . . . . . . . . . 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 3563 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘1)) → if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ)
163 nfcv 2319 . . . . . . . . . . . . . . . . 17 𝑛𝑥
164 nfv 1528 . . . . . . . . . . . . . . . . . 18 𝑛 𝑥𝑀
165164, 156, 147nfif 3562 . . . . . . . . . . . . . . . . 17 𝑛if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0)
16636, 158ifbieq1d 3556 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑥 → if(𝑛𝑀, 𝐶, 0) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
167163, 165, 166, 150fvmptf 5604 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℕ ∧ if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
1687, 162, 167syl2anc 411 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) = if(𝑥𝑀, 𝑥 / 𝑛𝐶, 0))
169168, 162eqeltrd 2254 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))‘𝑥) ∈ ℂ)
170 addcl 7927 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
171170adantl 277 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
17299, 154, 41, 169, 171seq3fveq 10457 . . . . . . . . . . . . 13 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀))
17312, 172jca 306 . . . . . . . . . . . 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 5445 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
175 fveq1 5510 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
176175csbeq1d 3064 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
177 vex 2740 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 ∈ V
178 vex 2740 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛 ∈ V
179177, 178fvex 5531 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑛) ∈ V
180175, 179eqeltrrdi 2269 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝐹 → (𝐹𝑛) ∈ V)
18110adantl 277 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 = 𝐹𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
182180, 181csbied 3103 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝐹(𝐹𝑛) / 𝑘𝐵 = 𝐶)
183176, 182eqtrd 2210 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = 𝐶)
184183ifeq1d 3551 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝐹 → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, 𝐶, 0))
185184mpteq2dv 4091 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))
186185seqeq3d 10439 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0))))
187186fveq1d 5513 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, 𝐶, 0)))‘𝑀))
188187eqeq2d 2189 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 2825 . . . . . . . . . . . 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 5877 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
193 f1oeq2 5446 . . . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑀 → (𝑛𝑚𝑛𝑀))
196195ifbid 3555 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑀 → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
197196mpteq2dv 4091 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑀 → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))
198197seqeq3d 10439 . . . . . . . . . . . . . . . 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 5518 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑀))
201200eqeq2d 2189 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 1825 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 411 . . . . . . . . . 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 734 . . . . . . . . 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 276 . . . . . . . 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 4004 . . . . . . . . . . . 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 1318 . . . . . . . . . . 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 2478 . . . . . . . . . 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 2184 . . . . . . . . . . . . 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 464 . . . . . . . . . . . 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 1825 . . . . . . . . . . 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 2478 . . . . . . . . . 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 793 . . . . . . . . 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 2918 . . . . . . . 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 1239 . . . . . . 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 115 . . . . . 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 157 . . . . . 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 129 . . . . 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 276 . . . 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 5194 . . 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 421 . 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, 223eqtrid 2222 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 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  ∃*wmo 2027  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  csb 3057  wss 3129  ifcif 3534   class class class wbr 4000  cmpt 4061  cio 5172  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  Fincfn 6734  cc 7800  0cc0 7802  1c1 7803   + caddc 7805  cle 7983  cn 8908  0cn0 9165  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739  cli 11270  Σcsu 11345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346
This theorem is referenced by:  isumz  11381  fsumf1o  11382  fsumcl2lem  11390  fsumadd  11398  sumsnf  11401  fsummulc2  11440
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