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Mirrors > Home > ILE Home > Th. List > grpmnd | GIF version |
Description: A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grpmnd | ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2177 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2177 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 1, 2, 3 | isgrp 12815 | . 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ (Base‘𝐺)∃𝑚 ∈ (Base‘𝐺)(𝑚(+g‘𝐺)𝑎) = (0g‘𝐺))) |
5 | 4 | simplbi 274 | 1 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 +gcplusg 12528 0gc0g 12693 Mndcmnd 12749 Grpcgrp 12809 |
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-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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-iota 5177 df-fv 5223 df-ov 5875 df-grp 12812 |
This theorem is referenced by: grpcl 12817 grpass 12818 grpideu 12820 grpmndd 12821 grpplusf 12823 grpplusfo 12824 grpsgrp 12833 dfgrp2 12834 grpidcl 12836 grplid 12838 grprid 12839 dfgrp3m 12901 mulgaddcom 12938 mulginvcom 12939 mulgz 12942 mulgneg2 12948 mulgass 12951 issubg3 12983 grpissubg 12985 0subg 12990 isabl2 13028 |
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