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Theorem zprod 15969
Description: Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)
Hypotheses
Ref Expression
zprod.1 𝑍 = (ℤ𝑀)
zprod.2 (𝜑𝑀 ∈ ℤ)
zprod.3 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
zprod.4 (𝜑𝐴𝑍)
zprod.5 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))
zprod.6 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
zprod (𝜑 → ∏𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))
Distinct variable groups:   𝐴,𝑘,𝑛,𝑦   𝐵,𝑛,𝑦   𝑘,𝐹   𝑘,𝑛,𝜑,𝑦   𝑘,𝑀,𝑦   𝜑,𝑛,𝑦   𝑛,𝑍
Allowed substitution hints:   𝐵(𝑘)   𝐹(𝑦,𝑛)   𝑀(𝑛)   𝑍(𝑦,𝑘)

Proof of Theorem zprod
Dummy variables 𝑓 𝑔 𝑖 𝑗 𝑚 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 1148 . . . . . . . 8 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
2 nfcv 2902 . . . . . . . . . . . 12 𝑖if(𝑘𝐴, 𝐵, 1)
3 nfv 1911 . . . . . . . . . . . . 13 𝑘 𝑖𝐴
4 nfcsb1v 3932 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵
5 nfcv 2902 . . . . . . . . . . . . 13 𝑘1
63, 4, 5nfif 4560 . . . . . . . . . . . 12 𝑘if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1)
7 eleq1w 2821 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝑘𝐴𝑖𝐴))
8 csbeq1a 3921 . . . . . . . . . . . . 13 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
97, 8ifbieq1d 4554 . . . . . . . . . . . 12 (𝑘 = 𝑖 → if(𝑘𝐴, 𝐵, 1) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1))
102, 6, 9cbvmpt 5258 . . . . . . . . . . 11 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1))
11 simpll 767 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝜑)
12 zprod.6 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1312ralrimiva 3143 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
144nfel1 2919 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
158eleq1d 2823 . . . . . . . . . . . . . 14 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
1614, 15rspc 3609 . . . . . . . . . . . . 13 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
1713, 16syl5 34 . . . . . . . . . . . 12 (𝑖𝐴 → (𝜑𝑖 / 𝑘𝐵 ∈ ℂ))
1811, 17mpan9 506 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
19 simplr 769 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
20 zprod.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
2120ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
22 simpr 484 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
23 zprod.4 . . . . . . . . . . . . 13 (𝜑𝐴𝑍)
24 zprod.1 . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
2523, 24sseqtrdi 4045 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ (ℤ𝑀))
2625ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
2710, 18, 19, 21, 22, 26prodrb 15964 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
2827biimpd 229 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
2928expimpd 453 . . . . . . . 8 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
301, 29syl5 34 . . . . . . 7 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
3130rexlimdva 3152 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
32 uzssz 12896 . . . . . . . . . . . . . . . . 17 (ℤ𝑀) ⊆ ℤ
33 zssre 12617 . . . . . . . . . . . . . . . . 17 ℤ ⊆ ℝ
3432, 33sstri 4004 . . . . . . . . . . . . . . . 16 (ℤ𝑀) ⊆ ℝ
3524, 34eqsstri 4029 . . . . . . . . . . . . . . 15 𝑍 ⊆ ℝ
3623, 35sstrdi 4007 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℝ)
3736ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ⊆ ℝ)
38 ltso 11338 . . . . . . . . . . . . 13 < Or ℝ
39 soss 5616 . . . . . . . . . . . . 13 (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴))
4037, 38, 39mpisyl 21 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → < Or 𝐴)
41 fzfi 14009 . . . . . . . . . . . . 13 (1...𝑚) ∈ Fin
42 ovex 7463 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ V
4342f1oen 9011 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴 → (1...𝑚) ≈ 𝐴)
4443adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
4544ensymd 9043 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
46 enfii 9223 . . . . . . . . . . . . 13 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
4741, 45, 46sylancr 587 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ∈ Fin)
48 fz1iso 14497 . . . . . . . . . . . 12 (( < Or 𝐴𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
4940, 47, 48syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
50 simpll 767 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
5150, 17mpan9 506 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
52 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
5352csbeq1d 3911 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
54 csbcow 3922 . . . . . . . . . . . . . . . 16 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
5553, 54eqtr4di 2792 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
5655cbvmptv 5260 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
57 eqid 2734 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵)
58 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
5920ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
6025ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
61 simprl 771 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
62 simprr 773 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
6310, 51, 56, 57, 58, 59, 60, 61, 62prodmolem2a 15966 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
6463expr 456 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6564exlimdv 1930 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6649, 65mpd 15 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
67 breq2 5151 . . . . . . . . . 10 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6866, 67syl5ibrcom 247 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
6968expimpd 453 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
7069exlimdv 1930 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
7170rexlimdva 3152 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
7231, 71jaod 859 . . . . 5 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
7320adantr 480 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
7423adantr 480 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝐴𝑍)
75 zprod.3 . . . . . . . . . 10 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
7624eleq2i 2830 . . . . . . . . . . . 12 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
77 eluzelz 12885 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
7877adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℤ)
79 uztrn 12893 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (ℤ𝑛) ∧ 𝑛 ∈ (ℤ𝑀)) → 𝑧 ∈ (ℤ𝑀))
8079ancoms 458 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (ℤ𝑛)) → 𝑧 ∈ (ℤ𝑀))
8124eleq2i 2830 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑍𝑘 ∈ (ℤ𝑀))
82 zprod.5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))
8324, 32eqsstri 4029 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑍 ⊆ ℤ
8483sseli 3990 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝑍𝑘 ∈ ℤ)
85 iftrue 4536 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8685adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8786, 12eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
8887ex 412 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ))
89 iffalse 4539 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 1)
90 ax-1cn 11210 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℂ
9189, 90eqeltrdi 2846 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
9288, 91pm2.61d1 180 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
93 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
9493fvmpt2 7026 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℤ ∧ if(𝑘𝐴, 𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = if(𝑘𝐴, 𝐵, 1))
9584, 92, 94syl2anr 597 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = if(𝑘𝐴, 𝐵, 1))
9682, 95eqtr4d 2777 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝑍) → (𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
9781, 96sylan2br 595 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
9897ralrimiva 3143 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
99 nffvmpt1 6917 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)
10099nfeq2 2920 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)
101 fveq2 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → (𝐹𝑘) = (𝐹𝑧))
102 fveq2 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
103101, 102eqeq12d 2750 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑧 → ((𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) ↔ (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)))
104100, 103rspc 3609 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)))
10598, 104mpan9 506 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ (ℤ𝑀)) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
10680, 105sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (ℤ𝑛))) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
107106anassrs 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑧 ∈ (ℤ𝑛)) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
10878, 107seqfeq 14064 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℤ𝑀)) → seq𝑛( · , 𝐹) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))))
109108breq1d 5157 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
110109anbi2d 630 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
111110exbidv 1918 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
11276, 111sylan2b 594 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
113112rexbidva 3174 . . . . . . . . . 10 (𝜑 → (∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
11475, 113mpbid 232 . . . . . . . . 9 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
115114adantr 480 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
116 simpr 484 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)
117 fveq2 6906 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
118117, 24eqtr4di 2792 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = 𝑍)
119118sseq2d 4027 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴𝑍))
120118rexeqdv 3324 . . . . . . . . . 10 (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
121 seqeq1 14041 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))))
122121breq1d 5157 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
123119, 120, 1223anbi123d 1435 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (𝐴𝑍 ∧ ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)))
124123rspcev 3621 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ (𝐴𝑍 ∧ ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
12573, 74, 115, 116, 124syl13anc 1371 . . . . . . 7 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
126125orcd 873 . . . . . 6 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
127126ex 412 . . . . 5 (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))))
12872, 127impbid 212 . . . 4 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
12995, 82eqtr4d 2777 . . . . . . . . 9 ((𝜑𝑘𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
13081, 129sylan2br 595 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
131130ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
13299nfeq1 2918 . . . . . . . 8 𝑘((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)
133102, 101eqeq12d 2750 . . . . . . . 8 (𝑘 = 𝑧 → (((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘) ↔ ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)))
134132, 133rspc 3609 . . . . . . 7 (𝑧 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)))
135131, 134mpan9 506 . . . . . 6 ((𝜑𝑧 ∈ (ℤ𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧))
13620, 135seqfeq 14064 . . . . 5 (𝜑 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑀( · , 𝐹))
137136breq1d 5157 . . . 4 (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
138128, 137bitrd 279 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
139138iotabidv 6546 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥))
140 df-prod 15936 . 2 𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
141 df-fv 6570 . 2 ( ⇝ ‘seq𝑀( · , 𝐹)) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥)
142139, 140, 1413eqtr4g 2799 1 (𝜑 → ∏𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  csb 3907  wss 3962  ifcif 4530   class class class wbr 5147  cmpt 5230   Or wor 5595  cio 6513  1-1-ontowf1o 6561  cfv 6562   Isom wiso 6563  (class class class)co 7430  cen 8980  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   · cmul 11157   < clt 11292  cn 12263  cz 12610  cuz 12875  ...cfz 13543  seqcseq 14038  chash 14365  cli 15516  cprod 15935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-prod 15936
This theorem is referenced by:  iprod  15970  zprodn0  15971  prodss  15979
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