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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 18112 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | mndass 17922 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
5 | 1, 4 | sylan 582 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Mndcmnd 17913 Grpcgrp 18105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-sgrp 17903 df-mnd 17914 df-grp 18108 |
This theorem is referenced by: grprcan 18139 grprinv 18155 grpinvid1 18156 grpinvid2 18157 grplcan 18163 grpasscan1 18164 grpasscan2 18165 grplmulf1o 18175 grpinvadd 18179 grpsubadd 18189 grpaddsubass 18191 grpsubsub4 18194 dfgrp3 18200 grplactcnv 18204 imasgrp 18217 mulgaddcomlem 18252 mulgaddcom 18253 mulgdirlem 18260 issubg2 18296 isnsg3 18314 nmzsubg 18319 ssnmz 18320 eqger 18332 eqgcpbl 18336 qusgrp 18337 conjghm 18391 conjnmz 18394 subgga 18432 cntzsubg 18469 sylow1lem2 18726 sylow2blem1 18747 sylow2blem2 18748 sylow2blem3 18749 sylow3lem1 18754 sylow3lem2 18755 lsmass 18797 lsmmod 18803 lsmdisj2 18810 gex2abl 18973 ringcom 19331 lmodass 19651 psrgrp 20180 evpmodpmf1o 20742 ghmcnp 22725 qustgpopn 22730 cnncvsaddassdemo 23769 ogrpaddltbi 30721 ogrpaddltrbid 30723 ogrpinvlt 30726 cyc3genpmlem 30795 archiabllem2c 30826 lfladdass 36211 dvhvaddass 38235 |
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