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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 21843. (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 2728 | . . . . 5 β’ (BaseβπΊ) = (BaseβπΊ) | |
3 | 1, 2 | zlmbas 21451 | . . . 4 β’ (BaseβπΊ) = (Baseβπ) |
4 | 3 | a1i 11 | . . 3 β’ (πΊ β Abel β (BaseβπΊ) = (Baseβπ)) |
5 | eqid 2728 | . . . . 5 β’ (+gβπΊ) = (+gβπΊ) | |
6 | 1, 5 | zlmplusg 21453 | . . . 4 β’ (+gβπΊ) = (+gβπ) |
7 | 6 | a1i 11 | . . 3 β’ (πΊ β Abel β (+gβπΊ) = (+gβπ)) |
8 | 1 | zlmsca 21457 | . . 3 β’ (πΊ β Abel β β€ring = (Scalarβπ)) |
9 | eqid 2728 | . . . . 5 β’ (.gβπΊ) = (.gβπΊ) | |
10 | 1, 9 | zlmvsca 21458 | . . . 4 β’ (.gβπΊ) = ( Β·π βπ) |
11 | 10 | a1i 11 | . . 3 β’ (πΊ β Abel β (.gβπΊ) = ( Β·π βπ)) |
12 | zringbas 21386 | . . . 4 β’ β€ = (Baseββ€ring) | |
13 | 12 | a1i 11 | . . 3 β’ (πΊ β Abel β β€ = (Baseββ€ring)) |
14 | zringplusg 21387 | . . . 4 β’ + = (+gββ€ring) | |
15 | 14 | a1i 11 | . . 3 β’ (πΊ β Abel β + = (+gββ€ring)) |
16 | zringmulr 21390 | . . . 4 β’ Β· = (.rββ€ring) | |
17 | 16 | a1i 11 | . . 3 β’ (πΊ β Abel β Β· = (.rββ€ring)) |
18 | zring1 21392 | . . . 4 β’ 1 = (1rββ€ring) | |
19 | 18 | a1i 11 | . . 3 β’ (πΊ β Abel β 1 = (1rββ€ring)) |
20 | zringring 21382 | . . . 4 β’ β€ring β Ring | |
21 | 20 | a1i 11 | . . 3 β’ (πΊ β Abel β β€ring β Ring) |
22 | 3, 6 | ablprop 19755 | . . . 4 β’ (πΊ β Abel β π β Abel) |
23 | ablgrp 19747 | . . . 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 19747 | . . . 4 β’ (πΊ β Abel β πΊ β Grp) | |
26 | 2, 9 | mulgcl 19053 | . . . 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 19788 | . . 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 19068 | . . . 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 578 | . . 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 19073 | . . . 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 578 | . . 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 19043 | . . . 4 β’ (π₯ β (BaseβπΊ) β (1(.gβπΊ)π₯) = π₯) |
34 | 33 | adantl 480 | . . 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 20756 | . 2 β’ (πΊ β Abel β π β LMod) |
36 | lmodabl 20799 | . . 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 6553 (class class class)co 7426 1c1 11147 + caddc 11149 Β· cmul 11151 β€cz 12596 Basecbs 17187 +gcplusg 17240 .rcmulr 17241 Β·π cvsca 17244 Grpcgrp 18897 .gcmg 19030 Abelcabl 19743 1rcur 20128 Ringcrg 20180 LModclmod 20750 β€ringczring 21379 β€Modczlm 21433 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-mulg 19031 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-cnfld 21287 df-zring 21380 df-zlm 21437 |
This theorem is referenced by: zlmassa 21843 zlmclm 25059 nmmulg 33602 cnzh 33604 rezh 33605 |
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