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Mirrors > Home > ILE Home > Th. List > sgrp0 | GIF version |
Description: Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.) |
Ref | Expression |
---|---|
sgrp0 | ⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgm0 12680 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm) | |
2 | rzal 3520 | . . 3 ⊢ ((Base‘𝑀) = ∅ → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))) | |
3 | 2 | adantl 277 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))) |
4 | eqid 2177 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
5 | eqid 2177 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
6 | 4, 5 | issgrp 12701 | . 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))) |
7 | 1, 3, 6 | sylanbrc 417 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∅c0 3422 ‘cfv 5212 (class class class)co 5869 Basecbs 12445 +gcplusg 12518 Mgmcmgm 12665 Smgrpcsgrp 12699 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-cnex 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-ov 5872 df-inn 8909 df-2 8967 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-mgm 12667 df-sgrp 12700 |
This theorem is referenced by: (None) |
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