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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 18750 | . 2 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm) | |
2 | sgrpass.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | sgrpass.o | . . 3 ⊢ ⚬ = (+g‘𝐺) | |
4 | 2, 3 | mgmcl 18669 | . 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1162 | 1 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Mgmcmgm 18664 Smgrpcsgrp 18744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-mgm 18666 df-sgrp 18745 |
This theorem is referenced by: sgrppropd 18757 prdsplusgsgrpcl 18758 cntzsgrpcl 19365 rngpropd 20192 |
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