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Theorem fprodseq 11575
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 11543 . 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 9552 . . . . 5 ℕ = (ℤ‘1)
3 1zzd 9269 . . . . 5 (𝜑 → 1 ∈ ℤ)
4 eqid 2177 . . . . . . 7 (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))
5 breq1 4003 . . . . . . . 8 (𝑛 = 𝑝 → (𝑛𝑀𝑝𝑀))
6 fveq2 5511 . . . . . . . 8 (𝑛 = 𝑝 → (𝐺𝑛) = (𝐺𝑝))
75, 6ifbieq1d 3556 . . . . . . 7 (𝑛 = 𝑝 → if(𝑛𝑀, (𝐺𝑛), 1) = if(𝑝𝑀, (𝐺𝑝), 1))
8 simpr 110 . . . . . . 7 ((𝜑𝑝 ∈ ℕ) → 𝑝 ∈ ℕ)
9 simpll 527 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → 𝜑)
108anim1i 340 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → (𝑝 ∈ ℕ ∧ 𝑝𝑀))
11 fprod.2 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℕ)
1211nnzd 9363 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
13 fznn 10075 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → (𝑝 ∈ (1...𝑀) ↔ (𝑝 ∈ ℕ ∧ 𝑝𝑀)))
1412, 13syl 14 . . . . . . . . . . 11 (𝜑 → (𝑝 ∈ (1...𝑀) ↔ (𝑝 ∈ ℕ ∧ 𝑝𝑀)))
1514ad2antrr 488 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → (𝑝 ∈ (1...𝑀) ↔ (𝑝 ∈ ℕ ∧ 𝑝𝑀)))
1610, 15mpbird 167 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → 𝑝 ∈ (1...𝑀))
176eleq1d 2246 . . . . . . . . . 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 11378 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
2322adantr 276 . . . . . . . . . 10 ((𝜑𝑝 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
24 simpr 110 . . . . . . . . . 10 ((𝜑𝑝 ∈ (1...𝑀)) → 𝑝 ∈ (1...𝑀))
2517, 23, 24rspcdva 2846 . . . . . . . . 9 ((𝜑𝑝 ∈ (1...𝑀)) → (𝐺𝑝) ∈ ℂ)
269, 16, 25syl2anc 411 . . . . . . . 8 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝𝑀) → (𝐺𝑝) ∈ ℂ)
27 1cnd 7964 . . . . . . . 8 (((𝜑𝑝 ∈ ℕ) ∧ ¬ 𝑝𝑀) → 1 ∈ ℂ)
288nnzd 9363 . . . . . . . . 9 ((𝜑𝑝 ∈ ℕ) → 𝑝 ∈ ℤ)
2912adantr 276 . . . . . . . . 9 ((𝜑𝑝 ∈ ℕ) → 𝑀 ∈ ℤ)
30 zdcle 9318 . . . . . . . . 9 ((𝑝 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑝𝑀)
3128, 29, 30syl2anc 411 . . . . . . . 8 ((𝜑𝑝 ∈ ℕ) → DECID 𝑝𝑀)
3226, 27, 31ifcldadc 3563 . . . . . . 7 ((𝜑𝑝 ∈ ℕ) → if(𝑝𝑀, (𝐺𝑝), 1) ∈ ℂ)
334, 7, 8, 32fvmptd3 5605 . . . . . 6 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑝) = if(𝑝𝑀, (𝐺𝑝), 1))
3433, 32eqeltrd 2254 . . . . 5 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑝) ∈ ℂ)
352, 3, 34prodf 11530 . . . 4 (𝜑 → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))):ℕ⟶ℂ)
3635, 11ffvelcdmd 5648 . . 3 (𝜑 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) ∈ ℂ)
37 eleq1w 2238 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖𝐴𝑗𝐴))
3837dcbid 838 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (DECID 𝑖𝐴DECID 𝑗𝐴))
3938cbvralv 2703 . . . . . . . . . . 11 (∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
4039anbi2i 457 . . . . . . . . . 10 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑖 ∈ (ℤ𝑚)DECID 𝑖𝐴) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴))
4140anbi1i 458 . . . . . . . . 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 2484 . . . . . . . 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 9172 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
44 hashfz1 10747 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
4543, 44syl 14 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (♯‘(1...𝑚)) = 𝑚)
4645adantr 276 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = 𝑚)
47 1zzd 9269 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 1 ∈ ℤ)
48 nnz 9261 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
4948adantr 276 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℤ)
5047, 49fzfigd 10417 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ∈ Fin)
51 simpr 110 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑓:(1...𝑚)–1-1-onto𝐴)
5250, 51fihasheqf1od 10753 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (♯‘(1...𝑚)) = (♯‘𝐴))
5346, 52eqtr3d 2212 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 = (♯‘𝐴))
5453breq2d 4012 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛𝑚𝑛 ≤ (♯‘𝐴)))
5554ifbid 3555 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1))
5655mpteq2dv 4091 . . . . . . . . . . . . . . 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 10439 . . . . . . . . . . . . . 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 5513 . . . . . . . . . . . . 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 2189 . . . . . . . . . . . 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 452 . . . . . . . . . . 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 1825 . . . . . . . . . 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 2492 . . . . . . . . 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 132 . . . . . . . 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 764 . . . . . . 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 5457 . . . . . . . . . . . . 13 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
6619, 65syl 14 . . . . . . . . . . . 12 (𝜑𝐹:(1...𝑀)⟶𝐴)
673, 12fzfigd 10417 . . . . . . . . . . . 12 (𝜑 → (1...𝑀) ∈ Fin)
68 fex 5741 . . . . . . . . . . . 12 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin) → 𝐹 ∈ V)
6966, 67, 68syl2anc 411 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
7011, 2eleqtrdi 2270 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘1))
71 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑢 → (𝐹𝑛) = (𝐹𝑢))
7271csbeq1d 3064 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢(𝐹𝑛) / 𝑘𝐵 = (𝐹𝑢) / 𝑘𝐵)
73 fveq2 5511 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢 → (𝐺𝑛) = (𝐺𝑢))
7472, 73eqeq12d 2192 . . . . . . . . . . . . . . 15 (𝑛 = 𝑢 → ((𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛) ↔ (𝐹𝑢) / 𝑘𝐵 = (𝐺𝑢)))
7566ffvelcdmda 5647 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
7618adantl 277 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
7775, 76csbied 3103 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
7877, 21eqtr4d 2213 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛))
7978ralrimiva 2550 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛))
8079adantr 276 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐹𝑛) / 𝑘𝐵 = (𝐺𝑛))
81 simpr 110 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢 ∈ (1...𝑀))
8274, 80, 81rspcdva 2846 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (1...𝑀)) → (𝐹𝑢) / 𝑘𝐵 = (𝐺𝑢))
83 eqid 2177 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))
84 breq1 4003 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴)))
8584, 72ifbieq1d 3556 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1) = if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1))
86 elfznn 10040 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (1...𝑀) → 𝑢 ∈ ℕ)
8786adantl 277 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢 ∈ ℕ)
88 elfzle2 10014 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ (1...𝑀) → 𝑢𝑀)
8988adantl 277 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢𝑀)
9011nnnn0d 9218 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℕ0)
91 hashfz1 10747 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
9290, 91syl 14 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
9367, 19fihasheqf1od 10753 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (♯‘(1...𝑀)) = (♯‘𝐴))
9492, 93eqtr3d 2212 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 = (♯‘𝐴))
9594adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑀 = (♯‘𝐴))
9689, 95breqtrd 4026 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢 ∈ (1...𝑀)) → 𝑢 ≤ (♯‘𝐴))
9796iftrued 3541 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1) = (𝐹𝑢) / 𝑘𝐵)
9897, 82eqtrd 2210 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1) = (𝐺𝑢))
9973eleq1d 2246 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑢 → ((𝐺𝑛) ∈ ℂ ↔ (𝐺𝑢) ∈ ℂ))
10022adantr 276 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ (1...𝑀)) → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
10199, 100, 81rspcdva 2846 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (1...𝑀)) → (𝐺𝑢) ∈ ℂ)
10298, 101eqeltrd 2254 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1) ∈ ℂ)
10383, 85, 87, 102fvmptd3 5605 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (𝐹𝑢) / 𝑘𝐵, 1))
104103, 97eqtrd 2210 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑢) = (𝐹𝑢) / 𝑘𝐵)
105 breq1 4003 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑢 → (𝑛𝑀𝑢𝑀))
106105, 73ifbieq1d 3556 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑢 → if(𝑛𝑀, (𝐺𝑛), 1) = if(𝑢𝑀, (𝐺𝑢), 1))
10789iftrued 3541 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢𝑀, (𝐺𝑢), 1) = (𝐺𝑢))
108107, 101eqeltrd 2254 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ (1...𝑀)) → if(𝑢𝑀, (𝐺𝑢), 1) ∈ ℂ)
1094, 106, 87, 108fvmptd3 5605 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑢) = if(𝑢𝑀, (𝐺𝑢), 1))
110109, 107eqtrd 2210 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑢) = (𝐺𝑢))
11182, 104, 1103eqtr4rd 2221 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑢) = ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑢))
112 elnnuz 9553 . . . . . . . . . . . . . 14 (𝑝 ∈ ℕ ↔ 𝑝 ∈ (ℤ‘1))
113112, 34sylan2br 288 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1))‘𝑝) ∈ ℂ)
114 breq1 4003 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑝 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑝 ≤ (♯‘𝐴)))
115 fveq2 5511 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑝 → (𝐹𝑛) = (𝐹𝑝))
116115csbeq1d 3064 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑝(𝐹𝑛) / 𝑘𝐵 = (𝐹𝑝) / 𝑘𝐵)
117114, 116ifbieq1d 3556 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1) = if(𝑝 ≤ (♯‘𝐴), (𝐹𝑝) / 𝑘𝐵, 1))
118 simpll 527 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝜑)
119 simpr 110 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ≤ (♯‘𝐴))
12094breq2d 4012 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑝𝑀𝑝 ≤ (♯‘𝐴)))
121120ad2antrr 488 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑝𝑀𝑝 ≤ (♯‘𝐴)))
122119, 121mpbird 167 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝𝑀)
123122, 16syldan 282 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (1...𝑀))
12466ffvelcdmda 5647 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (1...𝑀)) → (𝐹𝑝) ∈ 𝐴)
12520ralrimiva 2550 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
126125adantr 276 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
127 nfcsb1v 3090 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐹𝑝) / 𝑘𝐵
128127nfel1 2330 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐹𝑝) / 𝑘𝐵 ∈ ℂ
129 csbeq1a 3066 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝐹𝑝) → 𝐵 = (𝐹𝑝) / 𝑘𝐵)
130129eleq1d 2246 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝐹𝑝) → (𝐵 ∈ ℂ ↔ (𝐹𝑝) / 𝑘𝐵 ∈ ℂ))
131128, 130rspc 2835 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑝) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑝) / 𝑘𝐵 ∈ ℂ))
132124, 126, 131sylc 62 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (1...𝑀)) → (𝐹𝑝) / 𝑘𝐵 ∈ ℂ)
133118, 123, 132syl2anc 411 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ ℕ) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝐹𝑝) / 𝑘𝐵 ∈ ℂ)
134 1cnd 7964 . . . . . . . . . . . . . . . . 17 (((𝜑𝑝 ∈ ℕ) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
13594, 12eqeltrrd 2255 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘𝐴) ∈ ℤ)
136135adantr 276 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ ℕ) → (♯‘𝐴) ∈ ℤ)
137 zdcle 9318 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑝 ≤ (♯‘𝐴))
13828, 136, 137syl2anc 411 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ ℕ) → DECID 𝑝 ≤ (♯‘𝐴))
139133, 134, 138ifcldadc 3563 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ ℕ) → if(𝑝 ≤ (♯‘𝐴), (𝐹𝑝) / 𝑘𝐵, 1) ∈ ℂ)
14083, 117, 8, 139fvmptd3 5605 . . . . . . . . . . . . . . 15 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (𝐹𝑝) / 𝑘𝐵, 1))
141140, 139eqeltrd 2254 . . . . . . . . . . . . . 14 ((𝜑𝑝 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑝) ∈ ℂ)
142112, 141sylan2br 288 . . . . . . . . . . . . 13 ((𝜑𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))‘𝑝) ∈ ℂ)
143 mulcl 7929 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝 · 𝑞) ∈ ℂ)
144143adantl 277 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ)) → (𝑝 · 𝑞) ∈ ℂ)
14570, 111, 113, 142, 144seq3fveq 10457 . . . . . . . . . . . 12 (𝜑 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))‘𝑀))
14619, 145jca 306 . . . . . . . . . . 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 5445 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
148 fveq1 5510 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
149148csbeq1d 3064 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
150149ifeq1d 3551 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))
151150mpteq2dv 4091 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))
152151seqeq3d 10439 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1))))
153152fveq1d 5513 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝐹𝑛) / 𝑘𝐵, 1)))‘𝑀))
154153eqeq2d 2189 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 2826 . . . . . . . . . 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 5877 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
158157f1oeq2d 5453 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
159 fveq2 5511 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑀))
160159eqeq2d 2189 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 1825 . . . . . . . . . . 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 2841 . . . . . . . . . 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 411 . . . . . . . . 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 734 . . . . . . . 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 2319 . . . . . . . . . . . . . 14 𝑗if(𝑘𝐴, 𝐵, 1)
167 nfv 1528 . . . . . . . . . . . . . . 15 𝑘 𝑗𝐴
168 nfcsb1v 3090 . . . . . . . . . . . . . . 15 𝑘𝑗 / 𝑘𝐵
169 nfcv 2319 . . . . . . . . . . . . . . 15 𝑘1
170167, 168, 169nfif 3562 . . . . . . . . . . . . . 14 𝑘if(𝑗𝐴, 𝑗 / 𝑘𝐵, 1)
171 eleq1w 2238 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑘𝐴𝑗𝐴))
172 csbeq1a 3066 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
173171, 172ifbieq1d 3556 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 1) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 1))
174166, 170, 173cbvmpt 4095 . . . . . . . . . . . . 13 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 1))
175168nfel1 2330 . . . . . . . . . . . . . . 15 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
176172eleq1d 2246 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
177175, 176rspc 2835 . . . . . . . . . . . . . 14 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
178125, 177mpan9 281 . . . . . . . . . . . . 13 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
179 breq1 4003 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
180 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
181180csbeq1d 3064 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
182 csbcow 3068 . . . . . . . . . . . . . . . 16 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
183181, 182eqtr4di 2228 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
184179, 183ifbieq1d 3556 . . . . . . . . . . . . . 14 (𝑛 = 𝑖 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 1))
185184cbvmptv 4096 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 1)) = (𝑖 ∈ ℕ ↦ if(𝑖 ≤ (♯‘𝐴), (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵, 1))
186174, 178, 185prodmodc 11570 . . . . . . . . . . . 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 306 . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . 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 2184 . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . 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 1825 . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . 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 793 . . . . . . . . . . . 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 2918 . . . . . . . . . . 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 283 . . . . . . . . . 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 566 . . . . . . . . 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 375 . . . . . . . 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 421 . . . . . . 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, 201biimtrrid 153 . . . . . 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 194 . . . . . . 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 157 . . . . . 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 129 . . . . 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 276 . . . 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 5194 . . 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 421 . 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, 208eqtrid 2222 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 104  wb 105  wo 708  DECID wdc 834   = 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   · cmul 7807  cle 7983   # cap 8528  cn 8908  0cn0 9165  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739  cli 11270  cprod 11542
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-proddc 11543
This theorem is referenced by:  prod1dc  11578  fprodf1o  11580  fprodmul  11583  prodsnf  11584
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