|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sgrpcl | Structured version Visualization version GIF version | ||
| Description: Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| sgrpass.b | ⊢ 𝐵 = (Base‘𝐺) | 
| sgrpass.o | ⊢ ⚬ = (+g‘𝐺) | 
| Ref | Expression | 
|---|---|
| sgrpcl | ⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sgrpmgm 18737 | . 2 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm) | |
| 2 | sgrpass.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | sgrpass.o | . . 3 ⊢ ⚬ = (+g‘𝐺) | |
| 4 | 2, 3 | mgmcl 18656 | . 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | 
| 5 | 1, 4 | syl3an1 1164 | 1 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 Smgrpcsgrp 18731 | 
| 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-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-mgm 18653 df-sgrp 18732 | 
| This theorem is referenced by: sgrppropd 18744 prdsplusgsgrpcl 18745 cntzsgrpcl 19352 rngpropd 20171 | 
| Copyright terms: Public domain | W3C validator |