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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 21792. (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 2726 | . . . . 5 β’ (BaseβπΊ) = (BaseβπΊ) | |
3 | 1, 2 | zlmbas 21400 | . . . 4 β’ (BaseβπΊ) = (Baseβπ) |
4 | 3 | a1i 11 | . . 3 β’ (πΊ β Abel β (BaseβπΊ) = (Baseβπ)) |
5 | eqid 2726 | . . . . 5 β’ (+gβπΊ) = (+gβπΊ) | |
6 | 1, 5 | zlmplusg 21402 | . . . 4 β’ (+gβπΊ) = (+gβπ) |
7 | 6 | a1i 11 | . . 3 β’ (πΊ β Abel β (+gβπΊ) = (+gβπ)) |
8 | 1 | zlmsca 21406 | . . 3 β’ (πΊ β Abel β β€ring = (Scalarβπ)) |
9 | eqid 2726 | . . . . 5 β’ (.gβπΊ) = (.gβπΊ) | |
10 | 1, 9 | zlmvsca 21407 | . . . 4 β’ (.gβπΊ) = ( Β·π βπ) |
11 | 10 | a1i 11 | . . 3 β’ (πΊ β Abel β (.gβπΊ) = ( Β·π βπ)) |
12 | zringbas 21335 | . . . 4 β’ β€ = (Baseββ€ring) | |
13 | 12 | a1i 11 | . . 3 β’ (πΊ β Abel β β€ = (Baseββ€ring)) |
14 | zringplusg 21336 | . . . 4 β’ + = (+gββ€ring) | |
15 | 14 | a1i 11 | . . 3 β’ (πΊ β Abel β + = (+gββ€ring)) |
16 | zringmulr 21339 | . . . 4 β’ Β· = (.rββ€ring) | |
17 | 16 | a1i 11 | . . 3 β’ (πΊ β Abel β Β· = (.rββ€ring)) |
18 | zring1 21341 | . . . 4 β’ 1 = (1rββ€ring) | |
19 | 18 | a1i 11 | . . 3 β’ (πΊ β Abel β 1 = (1rββ€ring)) |
20 | zringring 21331 | . . . 4 β’ β€ring β Ring | |
21 | 20 | a1i 11 | . . 3 β’ (πΊ β Abel β β€ring β Ring) |
22 | 3, 6 | ablprop 19710 | . . . 4 β’ (πΊ β Abel β π β Abel) |
23 | ablgrp 19702 | . . . 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 19702 | . . . 4 β’ (πΊ β Abel β πΊ β Grp) | |
26 | 2, 9 | mulgcl 19015 | . . . 4 β’ ((πΊ β Grp β§ π₯ β β€ β§ π¦ β (BaseβπΊ)) β (π₯(.gβπΊ)π¦) β (BaseβπΊ)) |
27 | 25, 26 | syl3an1 1160 | . . 3 β’ ((πΊ β Abel β§ π₯ β β€ β§ π¦ β (BaseβπΊ)) β (π₯(.gβπΊ)π¦) β (BaseβπΊ)) |
28 | 2, 9, 5 | mulgdi 19743 | . . 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 19030 | . . . 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 579 | . . 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 19035 | . . . 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 579 | . . 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 19005 | . . . 4 β’ (π₯ β (BaseβπΊ) β (1(.gβπΊ)π₯) = π₯) |
34 | 33 | adantl 481 | . . 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 20709 | . 2 β’ (πΊ β Abel β π β LMod) |
36 | lmodabl 20752 | . . 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 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 1c1 11110 + caddc 11112 Β· cmul 11114 β€cz 12559 Basecbs 17150 +gcplusg 17203 .rcmulr 17204 Β·π cvsca 17207 Grpcgrp 18860 .gcmg 18992 Abelcabl 19698 1rcur 20083 Ringcrg 20135 LModclmod 20703 β€ringczring 21328 β€Modczlm 21382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-mulg 18993 df-subg 19047 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-subrng 20443 df-subrg 20468 df-lmod 20705 df-cnfld 21236 df-zring 21329 df-zlm 21386 |
This theorem is referenced by: zlmassa 21792 zlmclm 24989 nmmulg 33477 cnzh 33479 rezh 33480 |
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