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Mirrors > Home > MPE Home > Th. List > Mathboxes > drngringd | Structured version Visualization version GIF version |
Description: A division ring is a ring. (Contributed by SN, 16-May-2024.) |
Ref | Expression |
---|---|
drngringd.1 | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
Ref | Expression |
---|---|
drngringd | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngringd.1 | . 2 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
2 | drngring 19774 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Ringcrg 19562 DivRingcdr 19767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-drng 19769 |
This theorem is referenced by: drnggrpd 39960 drngmulcanad 39965 drngmulcan2ad 39966 drnginvmuld 39967 prjspner1 40171 |
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