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| Mirrors > Home > MPE Home > Th. List > Mathboxes > flddivrng | Structured version Visualization version GIF version | ||
| Description: A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| flddivrng | ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-fld 37999 | . . 3 ⊢ Fld = (DivRingOps ∩ Com2) | |
| 2 | inss1 4237 | . . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps | |
| 3 | 1, 2 | eqsstri 4030 | . 2 ⊢ Fld ⊆ DivRingOps | 
| 4 | 3 | sseli 3979 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∩ cin 3950 DivRingOpscdrng 37955 Com2ccm2 37996 Fldcfld 37998 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-ss 3968 df-fld 37999 | 
| This theorem is referenced by: isfld2 38012 isfldidl 38075 | 
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