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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 18958 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
| 2 | grpcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | grpcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 4 | 2, 3 | mndass 18756 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| 5 | 1, 4 | sylan 580 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mndcmnd 18747 Grpcgrp 18951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-sgrp 18732 df-mnd 18748 df-grp 18954 |
| This theorem is referenced by: grpassd 18963 grprcan 18991 grprinv 19008 grpinvid1 19009 grpinvid2 19010 grplcan 19018 grpasscan1 19019 grpasscan2 19020 grpinvadd 19036 grpsubadd 19046 grpaddsubass 19048 grpsubsub4 19051 dfgrp3 19057 grplactcnv 19061 imasgrp 19074 mulgaddcomlem 19115 mulgaddcom 19116 mulgdirlem 19123 issubg2 19159 isnsg3 19178 nmzsubg 19183 ssnmz 19184 eqgcpbl 19200 qusgrp 19204 conjghm 19267 subgga 19318 cntzsubg 19357 sylow1lem2 19617 sylow2blem1 19638 sylow2blem2 19639 sylow2blem3 19640 sylow3lem1 19645 sylow3lem2 19646 lsmass 19687 lsmmod 19693 lsmdisj2 19700 gex2abl 19869 ringcom 20277 lmodass 20874 evpmodpmf1o 21614 psrgrpOLD 21977 ghmcnp 24123 qustgpopn 24128 cnncvsaddassdemo 25197 ogrpaddltbi 33095 ogrpaddltrbid 33097 ogrpinvlt 33100 cyc3genpmlem 33171 archiabllem2c 33202 quslsm 33433 lfladdass 39074 dvhvaddass 41099 |
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