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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 15254 | . 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 6329 | . 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 2909 | 1 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 Vcvv 3494 ⦋csb 3882 ⊆ wss 3935 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 ℩cio 6306 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 · cmul 10536 ℕcn 11632 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 seqcseq 13363 ⇝ cli 14835 ∏cprod 15253 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 df-pr 4563 df-uni 4832 df-iota 6308 df-prod 15254 |
This theorem is referenced by: risefacval 15356 fallfacval 15357 prmoval 16363 fprodsubrecnncnvlem 42184 fprodaddrecnncnvlem 42186 etransclem13 42526 ovnlecvr 42834 ovncvrrp 42840 hoidmvval 42853 vonioolem1 42956 |
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