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Mirrors > Home > MPE Home > Th. List > drnggrp | Structured version Visualization version GIF version |
Description: A division ring is a group. (Contributed by NM, 8-Sep-2011.) |
Ref | Expression |
---|---|
drnggrp | ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 19512 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | ringgrp 19305 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 Grpcgrp 18106 Ringcrg 19300 DivRingcdr 19505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-ring 19302 df-drng 19507 |
This theorem is referenced by: drgextlsp 31000 qqh0 31229 qqhghm 31233 dvhvaddass 38237 dvhgrp 38247 cdlemn4 38338 |
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