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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 33791 | . . 3 ⊢ Fld = (DivRingOps ∩ Com2) | |
2 | inss1 3833 | . . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps | |
3 | 1, 2 | eqsstri 3635 | . 2 ⊢ Fld ⊆ DivRingOps |
4 | 3 | sseli 3599 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∩ cin 3573 DivRingOpscdrng 33747 Com2ccm2 33788 Fldcfld 33790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-fld 33791 |
This theorem is referenced by: isfld2 33804 isfldidl 33867 |
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