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Mirrors > Home > MPE Home > Th. List > Mathboxes > relrngo | Structured version Visualization version GIF version |
Description: The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
relrngo | ⊢ Rel RingOps |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rngo 37804 | . 2 ⊢ RingOps = {〈𝑔, ℎ〉 ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))} | |
2 | 1 | relopabiv 5843 | 1 ⊢ Rel RingOps |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ∀wral 3063 ∃wrex 3072 × cxp 5697 ran crn 5700 Rel wrel 5704 ⟶wf 6568 (class class class)co 7445 AbelOpcablo 30567 RingOpscrngo 37803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3484 df-ss 3987 df-opab 5232 df-xp 5705 df-rel 5706 df-rngo 37804 |
This theorem is referenced by: isrngo 37806 rngoi 37808 rngoablo2 37818 rngosn3 37833 isdrngo1 37865 iscrngo2 37906 |
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