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Mirrors > Home > MPE Home > Th. List > Mathboxes > drnggrpd | Structured version Visualization version GIF version |
Description: A division ring is a group. (Contributed by SN, 16-May-2024.) |
Ref | Expression |
---|---|
drngringd.1 | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
Ref | Expression |
---|---|
drnggrpd | ⊢ (𝜑 → 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngringd.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
2 | 1 | drngringd 39782 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
3 | 2 | ringgrpd 19379 | 1 ⊢ (𝜑 → 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Grpcgrp 18174 DivRingcdr 19575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-nul 5179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5036 df-iota 6298 df-fv 6347 df-ov 7158 df-ring 19372 df-drng 19577 |
This theorem is referenced by: (None) |
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