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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 36136 | . . 3 ⊢ Fld = (DivRingOps ∩ Com2) | |
2 | inss1 4163 | . . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps | |
3 | 1, 2 | eqsstri 3955 | . 2 ⊢ Fld ⊆ DivRingOps |
4 | 3 | sseli 3917 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3886 DivRingOpscdrng 36092 Com2ccm2 36133 Fldcfld 36135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3432 df-in 3894 df-ss 3904 df-fld 36136 |
This theorem is referenced by: isfld2 36149 isfldidl 36212 |
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