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| Mirrors > Home > MPE Home > Th. List > flddrngd | Structured version Visualization version GIF version | ||
| Description: A field is a division ring. (Contributed by SN, 17-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| flddrngd.1 | ⊢ (𝜑 → 𝑅 ∈ Field) | 
| Ref | Expression | 
|---|---|
| flddrngd | ⊢ (𝜑 → 𝑅 ∈ DivRing) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | flddrngd.1 | . 2 ⊢ (𝜑 → 𝑅 ∈ Field) | |
| 2 | isfld 20741 | . . 3 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) | |
| 3 | 2 | simplbi 497 | . 2 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing) | 
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → 𝑅 ∈ DivRing) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 CRingccrg 20232 DivRingcdr 20730 Fieldcfield 20731 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-in 3957 df-field 20733 | 
| This theorem is referenced by: ply1asclunit 33600 ply1unit 33601 ply1dg1rt 33605 m1pmeq 33609 fldgenfldext 33719 evls1fldgencl 33721 fldextrspunlsplem 33724 fldextrspunfld 33727 fldextrspunlem2 33728 fldextrspundgdvdslem 33731 fldextrspundgdvds 33732 minplyirred 33755 algextdeglem2 33760 algextdeglem3 33761 algextdeglem4 33762 algextdeglem5 33763 algextdeglem7 33765 algextdeglem8 33766 rtelextdg2lem 33768 rtelextdg2 33769 aks6d1c5lem3 42139 aks6d1c5lem2 42140 aks5lem7 42202 prjcrv0 42648 | 
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