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Mirrors > Home > MPE Home > Th. List > ringmgm | Structured version Visualization version GIF version |
Description: A ring is a magma. (Contributed by AV, 31-Jan-2020.) |
Ref | Expression |
---|---|
ringmgm | ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringmnd 19029 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
2 | mndmgm 17768 | . 2 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 Mgmcmgm 17708 Mndcmnd 17762 Ringcrg 19020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 ax-nul 5067 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-iota 6152 df-fv 6196 df-ov 6979 df-sgrp 17752 df-mnd 17763 df-grp 17894 df-ring 19022 |
This theorem is referenced by: gsumply1subr 20105 |
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