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Theorem zprod 15279
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 1141 . . . . . . . 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 2974 . . . . . . . . . . . 12 𝑖if(𝑘𝐴, 𝐵, 1)
3 nfv 1906 . . . . . . . . . . . . 13 𝑘 𝑖𝐴
4 nfcsb1v 3904 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵
5 nfcv 2974 . . . . . . . . . . . . 13 𝑘1
63, 4, 5nfif 4492 . . . . . . . . . . . 12 𝑘if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1)
7 eleq1w 2892 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝑘𝐴𝑖𝐴))
8 csbeq1a 3894 . . . . . . . . . . . . 13 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
97, 8ifbieq1d 4486 . . . . . . . . . . . 12 (𝑘 = 𝑖 → if(𝑘𝐴, 𝐵, 1) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1))
102, 6, 9cbvmpt 5158 . . . . . . . . . . 11 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1))
11 simpll 763 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝜑)
12 zprod.6 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1312ralrimiva 3179 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
144nfel1 2991 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
158eleq1d 2894 . . . . . . . . . . . . . 14 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
1614, 15rspc 3608 . . . . . . . . . . . . 13 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
1713, 16syl5 34 . . . . . . . . . . . 12 (𝑖𝐴 → (𝜑𝑖 / 𝑘𝐵 ∈ ℂ))
1811, 17mpan9 507 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
19 simplr 765 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
20 zprod.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
2120ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
22 simpr 485 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
23 zprod.4 . . . . . . . . . . . . 13 (𝜑𝐴𝑍)
24 zprod.1 . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
2523, 24sseqtrdi 4014 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ (ℤ𝑀))
2625ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
2710, 18, 19, 21, 22, 26prodrb 15274 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
2827biimpd 230 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
2928expimpd 454 . . . . . . . 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 3281 . . . . . 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 12252 . . . . . . . . . . . . . . . . 17 (ℤ𝑀) ⊆ ℤ
33 zssre 11976 . . . . . . . . . . . . . . . . 17 ℤ ⊆ ℝ
3432, 33sstri 3973 . . . . . . . . . . . . . . . 16 (ℤ𝑀) ⊆ ℝ
3524, 34eqsstri 3998 . . . . . . . . . . . . . . 15 𝑍 ⊆ ℝ
3623, 35sstrdi 3976 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℝ)
3736ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ⊆ ℝ)
38 ltso 10709 . . . . . . . . . . . . 13 < Or ℝ
39 soss 5486 . . . . . . . . . . . . 13 (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴))
4037, 38, 39mpisyl 21 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → < Or 𝐴)
41 fzfi 13328 . . . . . . . . . . . . 13 (1...𝑚) ∈ Fin
42 ovex 7178 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ V
4342f1oen 8518 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴 → (1...𝑚) ≈ 𝐴)
4443adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
4544ensymd 8548 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
46 enfii 8723 . . . . . . . . . . . . 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 13808 . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
5150, 17mpan9 507 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
52 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
5352csbeq1d 3884 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
54 csbcow 3895 . . . . . . . . . . . . . . . 16 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
5553, 54syl6eqr 2871 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
5655cbvmptv 5160 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
57 eqid 2818 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵)
58 simplr 765 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
5920ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
6025ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
61 simprl 767 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
62 simprr 769 . . . . . . . . . . . . . 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 15276 . . . . . . . . . . . . 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 457 . . . . . . . . . . . 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 1925 . . . . . . . . . . 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 5061 . . . . . . . . . 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 248 . . . . . . . . 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 454 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
7069exlimdv 1925 . . . . . . 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 3281 . . . . . 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 853 . . . . 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 481 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
7423adantr 481 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝐴𝑍)
75 zprod.3 . . . . . . . . . 10 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
7624eleq2i 2901 . . . . . . . . . . . 12 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
77 eluzelz 12241 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
7877adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℤ)
79 uztrn 12249 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (ℤ𝑛) ∧ 𝑛 ∈ (ℤ𝑀)) → 𝑧 ∈ (ℤ𝑀))
8079ancoms 459 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (ℤ𝑛)) → 𝑧 ∈ (ℤ𝑀))
8124eleq2i 2901 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑍𝑘 ∈ (ℤ𝑀))
82 zprod.5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))
8324, 32eqsstri 3998 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑍 ⊆ ℤ
8483sseli 3960 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝑍𝑘 ∈ ℤ)
85 iftrue 4469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8685adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8786, 12eqeltrd 2910 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
8887ex 413 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ))
89 iffalse 4472 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 1)
90 ax-1cn 10583 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℂ
9189, 90syl6eqel 2918 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
9288, 91pm2.61d1 181 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
93 eqid 2818 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
9493fvmpt2 6771 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℤ ∧ if(𝑘𝐴, 𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = if(𝑘𝐴, 𝐵, 1))
9584, 92, 94syl2anr 596 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = if(𝑘𝐴, 𝐵, 1))
9682, 95eqtr4d 2856 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝑍) → (𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
9781, 96sylan2br 594 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
9897ralrimiva 3179 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
99 nffvmpt1 6674 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)
10099nfeq2 2992 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)
101 fveq2 6663 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → (𝐹𝑘) = (𝐹𝑧))
102 fveq2 6663 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
103101, 102eqeq12d 2834 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑧 → ((𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) ↔ (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)))
104100, 103rspc 3608 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)))
10598, 104mpan9 507 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ (ℤ𝑀)) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
10680, 105sylan2 592 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (ℤ𝑛))) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
107106anassrs 468 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑧 ∈ (ℤ𝑛)) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
10878, 107seqfeq 13383 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℤ𝑀)) → seq𝑛( · , 𝐹) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))))
109108breq1d 5067 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
110109anbi2d 628 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
111110exbidv 1913 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
11276, 111sylan2b 593 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
113112rexbidva 3293 . . . . . . . . . 10 (𝜑 → (∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
11475, 113mpbid 233 . . . . . . . . 9 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
115114adantr 481 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
116 simpr 485 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)
117 fveq2 6663 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
118117, 24syl6eqr 2871 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = 𝑍)
119118sseq2d 3996 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴𝑍))
120118rexeqdv 3414 . . . . . . . . . 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 13360 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))))
122121breq1d 5067 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
123119, 120, 1223anbi123d 1427 . . . . . . . . 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 3620 . . . . . . . 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 1364 . . . . . . 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 869 . . . . . 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 413 . . . . 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 213 . . . 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 2856 . . . . . . . . 9 ((𝜑𝑘𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
13081, 129sylan2br 594 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
131130ralrimiva 3179 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
13299nfeq1 2990 . . . . . . . 8 𝑘((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)
133102, 101eqeq12d 2834 . . . . . . . 8 (𝑘 = 𝑧 → (((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘) ↔ ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)))
134132, 133rspc 3608 . . . . . . 7 (𝑧 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)))
135131, 134mpan9 507 . . . . . 6 ((𝜑𝑧 ∈ (ℤ𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧))
13620, 135seqfeq 13383 . . . . 5 (𝜑 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑀( · , 𝐹))
137136breq1d 5067 . . . 4 (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
138128, 137bitrd 280 . . 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 6332 . 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 15248 . 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 6356 . 2 ( ⇝ ‘seq𝑀( · , 𝐹)) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥)
142139, 140, 1413eqtr4g 2878 1 (𝜑 → ∏𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  csb 3880  wss 3933  ifcif 4463   class class class wbr 5057  cmpt 5137   Or wor 5466  cio 6305  1-1-ontowf1o 6347  cfv 6348   Isom wiso 6349  (class class class)co 7145  cen 8494  Fincfn 8497  cc 10523  cr 10524  0cc0 10525  1c1 10526   · cmul 10530   < clt 10663  cn 11626  cz 11969  cuz 12231  ...cfz 12880  seqcseq 13357  chash 13678  cli 14829  cprod 15247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-prod 15248
This theorem is referenced by:  iprod  15280  zprodn0  15281  prodss  15289
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