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Mirrors > Home > MPE Home > Th. List > grpass | Structured version Visualization version GIF version |
Description: A group operation is associative. (Contributed by NM, 14-Aug-2011.) |
Ref | Expression |
---|---|
grpcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpcl.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
grpass | ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 17252 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | mndass 17125 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
5 | 1, 4 | sylan 487 | 1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Mndcmnd 17117 Grpcgrp 17245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-sgrp 17107 df-mnd 17118 df-grp 17248 |
This theorem is referenced by: grprcan 17278 grprinv 17292 grpinvid1 17293 grpinvid2 17294 grplcan 17300 grpasscan1 17301 grpasscan2 17302 grplmulf1o 17312 grpinvadd 17316 grpsubadd 17326 grpaddsubass 17328 grpsubsub4 17331 dfgrp3 17337 grplactcnv 17341 imasgrp 17354 mulgaddcomlem 17386 mulgaddcom 17387 mulgdirlem 17395 issubg2 17432 isnsg3 17451 nmzsubg 17458 ssnmz 17459 eqger 17467 eqgcpbl 17471 qusgrp 17472 conjghm 17514 conjnmz 17517 subgga 17556 cntzsubg 17592 sylow1lem2 17837 sylow2blem1 17858 sylow2blem2 17859 sylow2blem3 17860 sylow3lem1 17865 sylow3lem2 17866 lsmass 17906 lsmmod 17911 lsmdisj2 17918 gex2abl 18077 ringcom 18402 lmodass 18701 psrgrp 19219 evpmodpmf1o 19761 ghmcnp 21728 qustgpopn 21733 cnncvsaddassdemo 22771 ogrpaddltbi 29050 ogrpaddltrbid 29052 ogrpinvlt 29055 archiabllem2c 29080 lfladdass 33378 dvhvaddass 35404 |
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