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Theorem zprod 14455
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 1051 . . . . . . . 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 2750 . . . . . . . . . . . 12 𝑖if(𝑘𝐴, 𝐵, 1)
3 nfv 1829 . . . . . . . . . . . . 13 𝑘 𝑖𝐴
4 nfcsb1v 3514 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵
5 nfcv 2750 . . . . . . . . . . . . 13 𝑘1
63, 4, 5nfif 4064 . . . . . . . . . . . 12 𝑘if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1)
7 eleq1 2675 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝑘𝐴𝑖𝐴))
8 csbeq1a 3507 . . . . . . . . . . . . 13 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
97, 8ifbieq1d 4058 . . . . . . . . . . . 12 (𝑘 = 𝑖 → if(𝑘𝐴, 𝐵, 1) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1))
102, 6, 9cbvmpt 4671 . . . . . . . . . . 11 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 1))
11 simpll 785 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝜑)
12 zprod.6 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1312ralrimiva 2948 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
144nfel1 2764 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
158eleq1d 2671 . . . . . . . . . . . . . 14 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
1614, 15rspc 3275 . . . . . . . . . . . . 13 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
1713, 16syl5 33 . . . . . . . . . . . 12 (𝑖𝐴 → (𝜑𝑖 / 𝑘𝐵 ∈ ℂ))
1811, 17mpan9 484 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
19 simplr 787 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
20 zprod.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
2120ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
22 simpr 475 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
23 zprod.4 . . . . . . . . . . . . 13 (𝜑𝐴𝑍)
24 zprod.1 . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
2523, 24syl6sseq 3613 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ (ℤ𝑀))
2625ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
2710, 18, 19, 21, 22, 26prodrb 14450 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
2827biimpd 217 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
2928expimpd 626 . . . . . . . 8 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
301, 29syl5 33 . . . . . . 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 3012 . . . . . 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 11542 . . . . . . . . . . . . . . . . 17 (ℤ𝑀) ⊆ ℤ
33 zssre 11220 . . . . . . . . . . . . . . . . 17 ℤ ⊆ ℝ
3432, 33sstri 3576 . . . . . . . . . . . . . . . 16 (ℤ𝑀) ⊆ ℝ
3524, 34eqsstri 3597 . . . . . . . . . . . . . . 15 𝑍 ⊆ ℝ
3623, 35syl6ss 3579 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℝ)
3736ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ⊆ ℝ)
38 ltso 9970 . . . . . . . . . . . . 13 < Or ℝ
39 soss 4967 . . . . . . . . . . . . 13 (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴))
4037, 38, 39mpisyl 21 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → < Or 𝐴)
41 fzfi 12591 . . . . . . . . . . . . 13 (1...𝑚) ∈ Fin
42 ovex 6555 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ V
4342f1oen 7840 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴 → (1...𝑚) ≈ 𝐴)
4443adantl 480 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
4544ensymd 7871 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
46 enfii 8040 . . . . . . . . . . . . 13 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
4741, 45, 46sylancr 693 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ∈ Fin)
48 fz1iso 13058 . . . . . . . . . . . 12 (( < Or 𝐴𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))
4940, 47, 48syl2anc 690 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))
50 simpll 785 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝜑)
5150, 17mpan9 484 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
52 fveq2 6088 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
5352csbeq1d 3505 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
54 csbco 3508 . . . . . . . . . . . . . . . 16 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
5553, 54syl6eqr 2661 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
5655cbvmptv 4672 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
57 eqid 2609 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵)
58 simplr 787 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
5920ad2antrr 757 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
6025ad2antrr 757 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
61 simprl 789 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
62 simprr 791 . . . . . . . . . . . . . 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 14452 . . . . . . . . . . . . 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 640 . . . . . . . . . . . 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 1847 . . . . . . . . . . 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 4581 . . . . . . . . . 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 235 . . . . . . . . 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 626 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
7069exlimdv 1847 . . . . . . 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 3012 . . . . . 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 393 . . . . 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 479 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
7423adantr 479 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝐴𝑍)
75 zprod.3 . . . . . . . . . 10 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
7624eleq2i 2679 . . . . . . . . . . . 12 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
77 eluzelz 11532 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
7877adantl 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℤ)
79 uztrn 11539 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (ℤ𝑛) ∧ 𝑛 ∈ (ℤ𝑀)) → 𝑧 ∈ (ℤ𝑀))
8079ancoms 467 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (ℤ𝑛)) → 𝑧 ∈ (ℤ𝑀))
8124eleq2i 2679 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑍𝑘 ∈ (ℤ𝑀))
82 zprod.5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))
8324, 32eqsstri 3597 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑍 ⊆ ℤ
8483sseli 3563 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝑍𝑘 ∈ ℤ)
85 iftrue 4041 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8685adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8786, 12eqeltrd 2687 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
8887ex 448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ))
89 iffalse 4044 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 1)
90 ax-1cn 9851 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℂ
9189, 90syl6eqel 2695 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
9288, 91pm2.61d1 169 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
93 eqid 2609 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
9493fvmpt2 6185 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℤ ∧ if(𝑘𝐴, 𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = if(𝑘𝐴, 𝐵, 1))
9584, 92, 94syl2anr 493 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = if(𝑘𝐴, 𝐵, 1))
9682, 95eqtr4d 2646 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝑍) → (𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
9781, 96sylan2br 491 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
9897ralrimiva 2948 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘))
99 nffvmpt1 6096 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)
10099nfeq2 2765 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)
101 fveq2 6088 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → (𝐹𝑘) = (𝐹𝑧))
102 fveq2 6088 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
103101, 102eqeq12d 2624 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑧 → ((𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) ↔ (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)))
104100, 103rspc 3275 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧)))
10598, 104mpan9 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧 ∈ (ℤ𝑀)) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
10680, 105sylan2 489 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (ℤ𝑛))) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
107106anassrs 677 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑧 ∈ (ℤ𝑛)) → (𝐹𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧))
10878, 107seqfeq 12646 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℤ𝑀)) → seq𝑛( · , 𝐹) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))))
109108breq1d 4587 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
110109anbi2d 735 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
111110exbidv 1836 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
11276, 111sylan2b 490 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
113112rexbidva 3030 . . . . . . . . . 10 (𝜑 → (∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)))
11475, 113mpbid 220 . . . . . . . . 9 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
115114adantr 479 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
116 simpr 475 . . . . . . . 8 ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)
117 fveq2 6088 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
118117, 24syl6eqr 2661 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = 𝑍)
119118sseq2d 3595 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴𝑍))
120118rexeqdv 3121 . . . . . . . . . 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 12624 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))))
122121breq1d 4587 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥))
123119, 120, 1223anbi123d 1390 . . . . . . . . 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 3281 . . . . . . . 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 1319 . . . . . . 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 405 . . . . . 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 448 . . . . 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 200 . . . 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 2646 . . . . . . . . 9 ((𝜑𝑘𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
13081, 129sylan2br 491 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
131130ralrimiva 2948 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘))
13299nfeq1 2763 . . . . . . . 8 𝑘((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)
133102, 101eqeq12d 2624 . . . . . . . 8 (𝑘 = 𝑧 → (((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘) ↔ ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)))
134132, 133rspc 3275 . . . . . . 7 (𝑧 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑘) = (𝐹𝑘) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧)))
135131, 134mpan9 484 . . . . . 6 ((𝜑𝑧 ∈ (ℤ𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑧) = (𝐹𝑧))
13620, 135seqfeq 12646 . . . . 5 (𝜑 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑀( · , 𝐹))
137136breq1d 4587 . . . 4 (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
138128, 137bitrd 266 . . 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 5775 . 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 14424 . 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 5798 . 2 ( ⇝ ‘seq𝑀( · , 𝐹)) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥)
142139, 140, 1413eqtr4g 2668 1 (𝜑 → ∏𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2779  wral 2895  wrex 2896  csb 3498  wss 3539  ifcif 4035   class class class wbr 4577  cmpt 4637   Or wor 4948  cio 5752  1-1-ontowf1o 5789  cfv 5790   Isom wiso 5791  (class class class)co 6527  cen 7816  Fincfn 7819  cc 9791  cr 9792  0cc0 9793  1c1 9794   · cmul 9798   < clt 9931  cn 10870  cz 11213  cuz 11522  ...cfz 12155  seqcseq 12621  #chash 12937  cli 14012  cprod 14423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-oi 8276  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-fz 12156  df-fzo 12293  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-clim 14016  df-prod 14424
This theorem is referenced by:  iprod  14456  zprodn0  14457  prodss  14465
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