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Mirrors > Home > MPE Home > Th. List > prodex | Structured version Visualization version GIF version |
Description: A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodex | ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-prod 15937 | . 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | |
2 | iotaex 6536 | . 2 ⊢ (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) ∈ V | |
3 | 1, 2 | eqeltri 2835 | 1 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 ⦋csb 3908 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ℩cio 6514 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 · cmul 11158 ℕcn 12264 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 seqcseq 14039 ⇝ cli 15517 ∏cprod 15936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-uni 4913 df-iota 6516 df-prod 15937 |
This theorem is referenced by: risefacval 16041 fallfacval 16042 prmoval 17067 fprodsubrecnncnvlem 45863 fprodaddrecnncnvlem 45865 etransclem13 46203 ovnlecvr 46514 ovncvrrp 46520 hoidmvval 46533 vonioolem1 46636 |
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