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Mirrors > Home > MPE Home > Th. List > mndsgrp | Structured version Visualization version GIF version |
Description: A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) |
Ref | Expression |
---|---|
mndsgrp | ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2736 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | ismnddef 18129 | . 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
4 | 3 | simplbi 501 | 1 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 Smgrpcsgrp 18116 Mndcmnd 18127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-mnd 18128 |
This theorem is referenced by: mndmgm 18134 mndass 18136 gsumccat 18222 mndsssgrp 18315 grpsgrp 18345 mulgnn0dir 18475 mulgnn0ass 18481 ringrng 45053 |
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