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Mirrors > Home > MPE Home > Th. List > ringgrpd | Structured version Visualization version GIF version |
Description: A ring is a group. (Contributed by SN, 16-May-2024.) |
Ref | Expression |
---|---|
ringgrpd.1 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
ringgrpd | ⊢ (𝜑 → 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrpd.1 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringgrp 19355 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 Grpcgrp 18154 Ringcrg 19350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-nul 5169 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-iota 6287 df-fv 6336 df-ov 7146 df-ring 19352 |
This theorem is referenced by: crnggrpd 19364 evlslem1 20830 znfermltl 31068 drnggrpd 39740 evlsbagval 39765 mhphf 39775 |
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