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Mirrors > Home > MPE Home > Th. List > zlmlmod | Structured version Visualization version GIF version |
Description: The β€-module operation turns an arbitrary abelian group into a left module over β€. Also see zlmassa 21321. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
zlmlmod.w | β’ π = (β€ModβπΊ) |
Ref | Expression |
---|---|
zlmlmod | β’ (πΊ β Abel β π β LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmlmod.w | . . . . 5 β’ π = (β€ModβπΊ) | |
2 | eqid 2737 | . . . . 5 β’ (BaseβπΊ) = (BaseβπΊ) | |
3 | 1, 2 | zlmbas 20935 | . . . 4 β’ (BaseβπΊ) = (Baseβπ) |
4 | 3 | a1i 11 | . . 3 β’ (πΊ β Abel β (BaseβπΊ) = (Baseβπ)) |
5 | eqid 2737 | . . . . 5 β’ (+gβπΊ) = (+gβπΊ) | |
6 | 1, 5 | zlmplusg 20937 | . . . 4 β’ (+gβπΊ) = (+gβπ) |
7 | 6 | a1i 11 | . . 3 β’ (πΊ β Abel β (+gβπΊ) = (+gβπ)) |
8 | 1 | zlmsca 20941 | . . 3 β’ (πΊ β Abel β β€ring = (Scalarβπ)) |
9 | eqid 2737 | . . . . 5 β’ (.gβπΊ) = (.gβπΊ) | |
10 | 1, 9 | zlmvsca 20942 | . . . 4 β’ (.gβπΊ) = ( Β·π βπ) |
11 | 10 | a1i 11 | . . 3 β’ (πΊ β Abel β (.gβπΊ) = ( Β·π βπ)) |
12 | zringbas 20891 | . . . 4 β’ β€ = (Baseββ€ring) | |
13 | 12 | a1i 11 | . . 3 β’ (πΊ β Abel β β€ = (Baseββ€ring)) |
14 | zringplusg 20892 | . . . 4 β’ + = (+gββ€ring) | |
15 | 14 | a1i 11 | . . 3 β’ (πΊ β Abel β + = (+gββ€ring)) |
16 | zringmulr 20894 | . . . 4 β’ Β· = (.rββ€ring) | |
17 | 16 | a1i 11 | . . 3 β’ (πΊ β Abel β Β· = (.rββ€ring)) |
18 | zring1 20896 | . . . 4 β’ 1 = (1rββ€ring) | |
19 | 18 | a1i 11 | . . 3 β’ (πΊ β Abel β 1 = (1rββ€ring)) |
20 | zringring 20888 | . . . 4 β’ β€ring β Ring | |
21 | 20 | a1i 11 | . . 3 β’ (πΊ β Abel β β€ring β Ring) |
22 | 3, 6 | ablprop 19582 | . . . 4 β’ (πΊ β Abel β π β Abel) |
23 | ablgrp 19574 | . . . 4 β’ (π β Abel β π β Grp) | |
24 | 22, 23 | sylbi 216 | . . 3 β’ (πΊ β Abel β π β Grp) |
25 | ablgrp 19574 | . . . 4 β’ (πΊ β Abel β πΊ β Grp) | |
26 | 2, 9 | mulgcl 18900 | . . . 4 β’ ((πΊ β Grp β§ π₯ β β€ β§ π¦ β (BaseβπΊ)) β (π₯(.gβπΊ)π¦) β (BaseβπΊ)) |
27 | 25, 26 | syl3an1 1164 | . . 3 β’ ((πΊ β Abel β§ π₯ β β€ β§ π¦ β (BaseβπΊ)) β (π₯(.gβπΊ)π¦) β (BaseβπΊ)) |
28 | 2, 9, 5 | mulgdi 19612 | . . 3 β’ ((πΊ β Abel β§ (π₯ β β€ β§ π¦ β (BaseβπΊ) β§ π§ β (BaseβπΊ))) β (π₯(.gβπΊ)(π¦(+gβπΊ)π§)) = ((π₯(.gβπΊ)π¦)(+gβπΊ)(π₯(.gβπΊ)π§))) |
29 | 2, 9, 5 | mulgdir 18915 | . . . 4 β’ ((πΊ β Grp β§ (π₯ β β€ β§ π¦ β β€ β§ π§ β (BaseβπΊ))) β ((π₯ + π¦)(.gβπΊ)π§) = ((π₯(.gβπΊ)π§)(+gβπΊ)(π¦(.gβπΊ)π§))) |
30 | 25, 29 | sylan 581 | . . 3 β’ ((πΊ β Abel β§ (π₯ β β€ β§ π¦ β β€ β§ π§ β (BaseβπΊ))) β ((π₯ + π¦)(.gβπΊ)π§) = ((π₯(.gβπΊ)π§)(+gβπΊ)(π¦(.gβπΊ)π§))) |
31 | 2, 9 | mulgass 18920 | . . . 4 β’ ((πΊ β Grp β§ (π₯ β β€ β§ π¦ β β€ β§ π§ β (BaseβπΊ))) β ((π₯ Β· π¦)(.gβπΊ)π§) = (π₯(.gβπΊ)(π¦(.gβπΊ)π§))) |
32 | 25, 31 | sylan 581 | . . 3 β’ ((πΊ β Abel β§ (π₯ β β€ β§ π¦ β β€ β§ π§ β (BaseβπΊ))) β ((π₯ Β· π¦)(.gβπΊ)π§) = (π₯(.gβπΊ)(π¦(.gβπΊ)π§))) |
33 | 2, 9 | mulg1 18890 | . . . 4 β’ (π₯ β (BaseβπΊ) β (1(.gβπΊ)π₯) = π₯) |
34 | 33 | adantl 483 | . . 3 β’ ((πΊ β Abel β§ π₯ β (BaseβπΊ)) β (1(.gβπΊ)π₯) = π₯) |
35 | 4, 7, 8, 11, 13, 15, 17, 19, 21, 24, 27, 28, 30, 32, 34 | islmodd 20344 | . 2 β’ (πΊ β Abel β π β LMod) |
36 | lmodabl 20385 | . . 3 β’ (π β LMod β π β Abel) | |
37 | 36, 22 | sylibr 233 | . 2 β’ (π β LMod β πΊ β Abel) |
38 | 35, 37 | impbii 208 | 1 β’ (πΊ β Abel β π β LMod) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6501 (class class class)co 7362 1c1 11059 + caddc 11061 Β· cmul 11063 β€cz 12506 Basecbs 17090 +gcplusg 17140 .rcmulr 17141 Β·π cvsca 17144 Grpcgrp 18755 .gcmg 18879 Abelcabl 19570 1rcur 19920 Ringcrg 19971 LModclmod 20338 β€ringczring 20885 β€Modczlm 20917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-fzo 13575 df-seq 13914 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-mulg 18880 df-subg 18932 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-subrg 20236 df-lmod 20340 df-cnfld 20813 df-zring 20886 df-zlm 20921 |
This theorem is referenced by: zlmassa 21321 zlmclm 24491 nmmulg 32589 cnzh 32591 rezh 32592 |
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