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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 32961 | . . 3 ⊢ Fld = (DivRingOps ∩ Com2) | |
2 | inss1 3795 | . . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps | |
3 | 1, 2 | eqsstri 3598 | . 2 ⊢ Fld ⊆ DivRingOps |
4 | 3 | sseli 3564 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∩ cin 3539 DivRingOpscdrng 32917 Com2ccm2 32958 Fldcfld 32960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-fld 32961 |
This theorem is referenced by: isfld2 32974 isfldidl 33037 |
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