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Theorem fprodseq 11457
 Description: The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
Hypotheses
Ref Expression
fprod.1 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
fprod.2 (𝜑𝑀 ∈ ℕ)
fprod.3 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
fprod.4 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fprod.5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
Assertion
Ref Expression
fprodseq (𝜑 → ∏𝑘𝐴 𝐵 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘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 fprodseq
Dummy variables 𝑓 𝑖 𝑗 𝑚 𝑥 𝑝 𝑞 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-proddc 11425 . 2 𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
2 nnuz 9453 . . . . 5 ℕ = (ℤ‘1)
3 1zzd 9173 . . . . 5 (𝜑 → 1 ∈ ℤ)
4 eqid 2154 . . . . . . 7 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))
5 breq1 3964 . . . . . . . 8 (𝑛 = 𝑝 → (𝑛𝑀𝑝𝑀))
6 fveq2 5461 . . . . . . . 8 (𝑛 = 𝑝 → (𝐺𝑛) = (𝐺𝑝))
75, 6ifbieq1d 3523 . . . . . . 7 (𝑛 = 𝑝 → if(𝑛𝑀, (𝐺𝑛), 1) = if(𝑝𝑀, (𝐺𝑝), 1))
8 simpr 109 . . . . . . 7 ((𝜑𝑝 ∈ ℕ) → 𝑝 ∈ ℕ)
9 simpll 519 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → 𝜑)
108anim1i 338 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → (𝑝 ∈ ℕ ∧ 𝑝𝑀))
11 fprod.2 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℕ)
1211nnzd 9264 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
13 fznn 9969 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → (𝑝 ∈ (1...𝑀) ↔ (𝑝 ∈ ℕ ∧ 𝑝𝑀)))
1412, 13syl 14 . . . . . . . . . . 11 (𝜑 → (𝑝 ∈ (1...𝑀) ↔ (𝑝 ∈ ℕ ∧ 𝑝𝑀)))
1514ad2antrr 480 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → (𝑝 ∈ (1...𝑀) ↔ (𝑝 ∈ ℕ ∧ 𝑝𝑀)))
1610, 15mpbird 166 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → 𝑝 ∈ (1...𝑀))
176eleq1d 2223 . . . . . . . . . 10 (𝑛 = 𝑝 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑝) ∈ ℂ))
18 fprod.1 . . . . . . . . . . . 12 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
19 fprod.3 . . . . . . . . . . . 12 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
20 fprod.4 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
21 fprod.5 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
2218, 11, 19, 20, 21fsumgcl 11260 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
2322adantr 274 . . . . . . . . . 10 ((𝜑𝑝 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
24 simpr 109 . . . . . . . . . 10 ((𝜑𝑝 ∈ (1...𝑀)) → 𝑝 ∈ (1...𝑀))
2517, 23, 24rspcdva 2818 . . . . . . . . 9 ((𝜑𝑝 ∈ (1...𝑀)) → (𝐺𝑝) ∈ ℂ)
269, 16, 25syl2anc 409 . . . . . . . 8 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → (𝐺𝑝) ∈ ℂ)
27 1cnd 7873 . . . . . . . 8 (((𝜑𝑝 ∈ ℕ) ∧ ¬ 𝑝𝑀) → 1 ∈ ℂ)
288nnzd 9264 . . . . . . . . 9 ((𝜑𝑝 ∈ ℕ) → 𝑝 ∈ ℤ)
2912adantr 274 . . . . . . . . 9 ((𝜑𝑝 ∈ ℕ) → 𝑀 ∈ ℤ)
30 zdcle 9219 . . . . . . . . 9 ((𝑝 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑝𝑀)
3128, 29, 30syl2anc 409 . . . . . . . 8 ((𝜑𝑝 ∈ ℕ) → DECID 𝑝𝑀)
3226, 27, 31ifcldadc 3530 . . . . . . 7 ((𝜑𝑝 ∈ ℕ) → if(𝑝𝑀, (𝐺𝑝), 1) ∈ ℂ)
334, 7, 8, 32fvmptd3 5554 . . . . . 6 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑝) = if(𝑝𝑀, (𝐺𝑝), 1))
3433, 32eqeltrd 2231 . . . . 5 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑝) ∈ ℂ)
352, 3, 34prodf 11412 . . . 4 (𝜑 → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))):ℕ⟶ℂ)
3635, 11ffvelrnd 5596 . . 3 (𝜑 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) ∈ ℂ)
37 eleq1w 2215 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖𝐴𝑗𝐴))
3837dcbid 824 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (DECID 𝑖𝐴DECID 𝑗𝐴))
3938cbvralv 2677 . . . . . . . . . . 11 (∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
4039anbi2i 453 . . . . . . . . . 10 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴))
4140anbi1i 454 . . . . . . . . 9 (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)))
4241rexbii 2461 . . . . . . . 8 (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)))
43 nnnn0 9076 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
44 hashfz1 10634 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
4543, 44syl 14 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (♯‘(1...𝑚)) = 𝑚)
4645adantr 274 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = 𝑚)
47 1zzd 9173 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 1 ∈ ℤ)
48 nnz 9165 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
4948adantr 274 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℤ)
5047, 49fzfigd 10308 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ∈ Fin)
51 simpr 109 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑓:(1...𝑚)–1-1-onto𝐴)
5250, 51fihasheqf1od 10641 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
5346, 52eqtr3d 2189 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 = (♯‘𝐴))
5453breq2d 3973 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛𝑚𝑛 ≤ (♯‘𝐴)))
5554ifbid 3522 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1))
5655mpteq2dv 4051 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))
5756seqeq3d 10330 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1))))
5857fveq1d 5463 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))
5958eqeq2d 2166 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))
6059pm5.32da 448 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
6160exbidv 1802 . . . . . . . . . 10 (𝑚 ∈ ℕ → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
6261rexbiia 2469 . . . . . . . . 9 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))
6362bicomi 131 . . . . . . . 8 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))
6442, 63orbi12i 754 . . . . . . 7 ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
65 f1of 5407 . . . . . . . . . . . . 13 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
6619, 65syl 14 . . . . . . . . . . . 12 (𝜑𝐹:(1...𝑀)⟶𝐴)
673, 12fzfigd 10308 . . . . . . . . . . . 12 (𝜑 → (1...𝑀) ∈ Fin)
68 fex 5687 . . . . . . . . . . . 12 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin) → 𝐹 ∈ V)
6966, 67, 68syl2anc 409 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
7011, 2eleqtrdi 2247 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘1))
71 fveq2 5461 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑢 → (𝐹𝑛) = (𝐹𝑢))
7271csbeq1d 3034 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢(𝐹𝑛) / 𝑘𝐵 = (𝐹𝑢) / 𝑘𝐵)
73 fveq2 5461 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢 → (𝐺𝑛) = (𝐺𝑢))
7472, 73eqeq12d 2169 . . . . . . . . . . . . . . 15 (𝑛 = 𝑢 → ((𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛) ↔ (𝐹𝑢) / 𝑘𝐵 = (𝐺𝑢)))
7566ffvelrnda 5595 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
7618adantl 275 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
7775, 76csbied 3073 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
7877, 21eqtr4d 2190 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛))
7978ralrimiva 2527 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛))
8079adantr 274 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛))
81 simpr 109 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢 ∈ (1...𝑀))
8274, 80, 81rspcdva 2818 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (1...𝑀)) → (𝐹𝑢) / 𝑘𝐵 = (𝐺𝑢))
83 eqid 2154 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))
84 breq1 3964 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴)))
8584, 72ifbieq1d 3523 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1) = if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1))
86 elfznn 9934 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (1...𝑀) → 𝑢 ∈ ℕ)
8786adantl 275 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢 ∈ ℕ)
88 elfzle2 9908 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ (1...𝑀) → 𝑢𝑀)
8988adantl 275 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢𝑀)
9011nnnn0d 9122 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℕ0)
91 hashfz1 10634 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
9290, 91syl 14 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
9367, 19fihasheqf1od 10641 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (♯‘(1...𝑀)) = (♯‘𝐴))
9492, 93eqtr3d 2189 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 = (♯‘𝐴))
9594adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑀 = (♯‘𝐴))
9689, 95breqtrd 3986 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢 ≤ (♯‘𝐴))
9796iftrued 3508 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1) = (𝐹𝑢) / 𝑘𝐵)
9897, 82eqtrd 2187 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1) = (𝐺𝑢))
9973eleq1d 2223 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑢 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑢) ∈ ℂ))
10022adantr 274 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
10199, 100, 81rspcdva 2818 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (1...𝑀)) → (𝐺𝑢) ∈ ℂ)
10298, 101eqeltrd 2231 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1) ∈ ℂ)
10383, 85, 87, 102fvmptd3 5554 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1))
104103, 97eqtrd 2187 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑢) = (𝐹𝑢) / 𝑘𝐵)
105 breq1 3964 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑢 → (𝑛𝑀𝑢𝑀))
106105, 73ifbieq1d 3523 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢 → if(𝑛𝑀, (𝐺𝑛), 1) = if(𝑢𝑀, (𝐺𝑢), 1))
10789iftrued 3508 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢𝑀, (𝐺𝑢), 1) = (𝐺𝑢))
108107, 101eqeltrd 2231 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢𝑀, (𝐺𝑢), 1) ∈ ℂ)
1094, 106, 87, 108fvmptd3 5554 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑢) = if(𝑢𝑀, (𝐺𝑢), 1))
110109, 107eqtrd 2187 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑢) = (𝐺𝑢))
11182, 104, 1103eqtr4rd 2198 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑢) = ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑢))
112 elnnuz 9454 . . . . . . . . . . . . . 14 (𝑝 ∈ ℕ ↔ 𝑝 ∈ (ℤ‘1))
113112, 34sylan2br 286 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑝) ∈ ℂ)
114 breq1 3964 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑝 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑝 ≤ (♯‘𝐴)))
115 fveq2 5461 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑝 → (𝐹𝑛) = (𝐹𝑝))
116115csbeq1d 3034 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑝(𝐹𝑛) / 𝑘𝐵 = (𝐹𝑝) / 𝑘𝐵)
117114, 116ifbieq1d 3523 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1) = if(𝑝 ≤ (♯‘𝐴), (𝐹𝑝) / 𝑘𝐵, 1))
118 simpll 519 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝜑)
119 simpr 109 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ≤ (♯‘𝐴))
12094breq2d 3973 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑝𝑀𝑝 ≤ (♯‘𝐴)))
121120ad2antrr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑝𝑀𝑝 ≤ (♯‘𝐴)))
122119, 121mpbird 166 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝𝑀)
123122, 16syldan 280 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (1...𝑀))
12466ffvelrnda 5595 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (1...𝑀)) → (𝐹𝑝) ∈ 𝐴)
12520ralrimiva 2527 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
126125adantr 274 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
127 nfcsb1v 3060 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐹𝑝) / 𝑘𝐵
128127nfel1 2307 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐹𝑝) / 𝑘𝐵 ∈ ℂ
129 csbeq1a 3036 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝐹𝑝) → 𝐵 = (𝐹𝑝) / 𝑘𝐵)
130129eleq1d 2223 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝐹𝑝) → (𝐵 ∈ ℂ ↔ (𝐹𝑝) / 𝑘𝐵 ∈ ℂ))
131128, 130rspc 2807 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑝) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑝) / 𝑘𝐵 ∈ ℂ))
132124, 126, 131sylc 62 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (1...𝑀)) → (𝐹𝑝) / 𝑘𝐵 ∈ ℂ)
133118, 123, 132syl2anc 409 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝐹𝑝) / 𝑘𝐵 ∈ ℂ)
134 1cnd 7873 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ ℕ) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
13594, 12eqeltrrd 2232 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘𝐴) ∈ ℤ)
136135adantr 274 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ ℕ) → (♯‘𝐴) ∈ ℤ)
137 zdcle 9219 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑝 ≤ (♯‘𝐴))
13828, 136, 137syl2anc 409 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ ℕ) → DECID 𝑝 ≤ (♯‘𝐴))
139133, 134, 138ifcldadc 3530 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ ℕ) → if(𝑝 ≤ (♯‘𝐴), (𝐹𝑝) / 𝑘𝐵, 1) ∈ ℂ)
14083, 117, 8, 139fvmptd3 5554 . . . . . . . . . . . . . . 15 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (𝐹𝑝) / 𝑘𝐵, 1))
141140, 139eqeltrd 2231 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑝) ∈ ℂ)
142112, 141sylan2br 286 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑝) ∈ ℂ)
143 mulcl 7838 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝 · 𝑞) ∈ ℂ)
144143adantl 275 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ)) → (𝑝 · 𝑞) ∈ ℂ)
14570, 111, 113, 142, 144seq3fveq 10348 . . . . . . . . . . . 12 (𝜑 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))‘𝑀))
14619, 145jca 304 . . . . . . . . . . 11 (𝜑 → (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))‘𝑀)))
147 f1oeq1 5396 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
148 fveq1 5460 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
149148csbeq1d 3034 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
150149ifeq1d 3518 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))
151150mpteq2dv 4051 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))
152151seqeq3d 10330 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))))
153152fveq1d 5463 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))‘𝑀))
154153eqeq2d 2166 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀) ↔ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))‘𝑀)))
155147, 154anbi12d 465 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀)) ↔ (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))‘𝑀))))
15669, 146, 155spcedv 2798 . . . . . . . . . 10 (𝜑 → ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀)))
157 oveq2 5822 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
158157f1oeq2d 5403 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
159 fveq2 5461 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀))
160159eqeq2d 2166 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) ↔ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀)))
161158, 160anbi12d 465 . . . . . . . . . . . 12 (𝑚 = 𝑀 → ((𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ (𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀))))
162161exbidv 1802 . . . . . . . . . . 11 (𝑚 = 𝑀 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀))))
163162rspcev 2813 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀))) → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))
16411, 156, 163syl2anc 409 . . . . . . . . 9 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))
165164olcd 724 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
166 nfcv 2296 . . . . . . . . . . . . . 14 𝑗if(𝑘𝐴, 𝐵, 1)
167 nfv 1505 . . . . . . . . . . . . . . 15 𝑘 𝑗𝐴
168 nfcsb1v 3060 . . . . . . . . . . . . . . 15 𝑘𝑗 / 𝑘𝐵
169 nfcv 2296 . . . . . . . . . . . . . . 15 𝑘1
170167, 168, 169nfif 3529 . . . . . . . . . . . . . 14 𝑘if(𝑗𝐴, 𝑗 / 𝑘𝐵, 1)
171 eleq1w 2215 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑘𝐴𝑗𝐴))
172 csbeq1a 3036 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
173171, 172ifbieq1d 3523 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 1) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 1))
174166, 170, 173cbvmpt 4055 . . . . . . . . . . . . 13 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 1))
175168nfel1 2307 . . . . . . . . . . . . . . 15 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
176172eleq1d 2223 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
177175, 176rspc 2807 . . . . . . . . . . . . . 14 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
178125, 177mpan9 279 . . . . . . . . . . . . 13 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
179 breq1 3964 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
180 fveq2 5461 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
181180csbeq1d 3034 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
182 csbcow 3038 . . . . . . . . . . . . . . . 16 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
183181, 182eqtr4di 2205 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
184179, 183ifbieq1d 3523 . . . . . . . . . . . . . 14 (𝑛 = 𝑖 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 1))
185184cbvmptv 4056 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)) = (𝑖 ∈ ℕ ↦ if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 1))
186174, 178, 185prodmodc 11452 . . . . . . . . . . . 12 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
18736, 186jca 304 . . . . . . . . . . 11 (𝜑 → ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) ∈ ℂ ∧ ∃*𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))))
188 breq2 3965 . . . . . . . . . . . . . . . 16 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))
189188anbi2d 460 . . . . . . . . . . . . . . 15 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → ((∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))))
190189anbi2d 460 . . . . . . . . . . . . . 14 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))))
191190rexbidv 2455 . . . . . . . . . . . . 13 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))))
192 eqeq1 2161 . . . . . . . . . . . . . . . 16 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) ↔ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))
193192anbi2d 460 . . . . . . . . . . . . . . 15 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
194193exbidv 1802 . . . . . . . . . . . . . 14 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
195194rexbidv 2455 . . . . . . . . . . . . 13 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
196191, 195orbi12d 783 . . . . . . . . . . . 12 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))))
197196moi2 2889 . . . . . . . . . . 11 ((((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) ∈ ℂ ∧ ∃*𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))) ∧ ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ∧ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))) → 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))
198187, 197sylan 281 . . . . . . . . . 10 ((𝜑 ∧ ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ∧ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))) → 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))
199198ancom2s 556 . . . . . . . . 9 ((𝜑 ∧ ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ∧ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))) → 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))
200199expr 373 . . . . . . . 8 ((𝜑 ∧ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))) → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) → 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))
201165, 200mpdan 418 . . . . . . 7 (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) → 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))
20264, 201syl5bir 152 . . . . . 6 (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) → 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))
20364, 196bitr3id 193 . . . . . . 7 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))))
204165, 203syl5ibrcom 156 . . . . . 6 (𝜑 → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))))
205202, 204impbid 128 . . . . 5 (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))
206205adantr 274 . . . 4 ((𝜑 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) ∈ ℂ) → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀)))
207206iota5 5148 . . 3 ((𝜑 ∧ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) ∈ ℂ) → (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))
20836, 207mpdan 418 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))
2091, 208syl5eq 2199 1 (𝜑 → ∏𝑘𝐴 𝐵 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820   = 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   · cmul 7716   ≤ cle 7892   # cap 8435  ℕcn 8812  ℕ0cn0 9069  ℤcz 9146  ℤ≥cuz 9418  ...cfz 9890  seqcseq 10322  ♯chash 10626   ⇝ cli 11152  ∏cprod 11424 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-proddc 11425 This theorem is referenced by:  prod1dc  11460  fprodf1o  11462  fprodmul  11465  prodsnf  11466
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