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Mirrors > Home > ILE Home > Th. List > mndmgm | GIF version |
Description: A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) |
Ref | Expression |
---|---|
mndmgm | ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndsgrp 12847 | . 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Smgrp) | |
2 | sgrpmgm 12835 | . 2 ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2159 Mgmcmgm 12795 Smgrpcsgrp 12829 Mndcmnd 12842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-cnex 7919 ax-resscn 7920 ax-1re 7922 ax-addrcl 7925 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-sbc 2977 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-iota 5192 df-fun 5232 df-fn 5233 df-fv 5238 df-ov 5893 df-inn 8937 df-2 8995 df-ndx 12482 df-slot 12483 df-base 12485 df-plusg 12567 df-sgrp 12830 df-mnd 12843 |
This theorem is referenced by: mndcl 12849 mndplusf 12859 mndissubm 12892 grpissubg 13098 srg1zr 13301 ringmgm 13321 |
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