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| Mirrors > Home > MPE Home > Th. List > grpsgrp | Structured version Visualization version GIF version | ||
| Description: A group is a semigroup. (Contributed by AV, 28-Aug-2021.) | 
| Ref | Expression | 
|---|---|
| grpsgrp | ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | grpmnd 18958 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
| 2 | mndsgrp 18753 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Smgrpcsgrp 18731 Mndcmnd 18747 Grpcgrp 18951 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-mnd 18748 df-grp 18954 | 
| This theorem is referenced by: dfgrp2 18980 dfgrp3 19057 isarchi3 33194 | 
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