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Mirrors > Home > MPE Home > Th. List > zlmassa | Structured version Visualization version GIF version |
Description: The β€-module operation turns a ring into an associative algebra over β€. Also see zlmlmod 20943. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
zlmassa.w | β’ π = (β€ModβπΊ) |
Ref | Expression |
---|---|
zlmassa | β’ (πΊ β Ring β π β AssAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmassa.w | . . . . 5 β’ π = (β€ModβπΊ) | |
2 | eqid 2733 | . . . . 5 β’ (BaseβπΊ) = (BaseβπΊ) | |
3 | 1, 2 | zlmbas 20935 | . . . 4 β’ (BaseβπΊ) = (Baseβπ) |
4 | 3 | a1i 11 | . . 3 β’ (πΊ β Ring β (BaseβπΊ) = (Baseβπ)) |
5 | 1 | zlmsca 20941 | . . 3 β’ (πΊ β Ring β β€ring = (Scalarβπ)) |
6 | zringbas 20891 | . . . 4 β’ β€ = (Baseββ€ring) | |
7 | 6 | a1i 11 | . . 3 β’ (πΊ β Ring β β€ = (Baseββ€ring)) |
8 | eqid 2733 | . . . . 5 β’ (.gβπΊ) = (.gβπΊ) | |
9 | 1, 8 | zlmvsca 20942 | . . . 4 β’ (.gβπΊ) = ( Β·π βπ) |
10 | 9 | a1i 11 | . . 3 β’ (πΊ β Ring β (.gβπΊ) = ( Β·π βπ)) |
11 | eqid 2733 | . . . . 5 β’ (.rβπΊ) = (.rβπΊ) | |
12 | 1, 11 | zlmmulr 20939 | . . . 4 β’ (.rβπΊ) = (.rβπ) |
13 | 12 | a1i 11 | . . 3 β’ (πΊ β Ring β (.rβπΊ) = (.rβπ)) |
14 | ringabl 20007 | . . . 4 β’ (πΊ β Ring β πΊ β Abel) | |
15 | 1 | zlmlmod 20943 | . . . 4 β’ (πΊ β Abel β π β LMod) |
16 | 14, 15 | sylib 217 | . . 3 β’ (πΊ β Ring β π β LMod) |
17 | eqid 2733 | . . . . . 6 β’ (+gβπΊ) = (+gβπΊ) | |
18 | 1, 17 | zlmplusg 20937 | . . . . 5 β’ (+gβπΊ) = (+gβπ) |
19 | 3, 18, 12 | ringprop 20013 | . . . 4 β’ (πΊ β Ring β π β Ring) |
20 | 19 | biimpi 215 | . . 3 β’ (πΊ β Ring β π β Ring) |
21 | zringcrng 20887 | . . . 4 β’ β€ring β CRing | |
22 | 21 | a1i 11 | . . 3 β’ (πΊ β Ring β β€ring β CRing) |
23 | 2, 8, 11 | mulgass2 20030 | . . 3 β’ ((πΊ β Ring β§ (π₯ β β€ β§ π¦ β (BaseβπΊ) β§ π§ β (BaseβπΊ))) β ((π₯(.gβπΊ)π¦)(.rβπΊ)π§) = (π₯(.gβπΊ)(π¦(.rβπΊ)π§))) |
24 | 2, 8, 11 | mulgass3 20071 | . . 3 β’ ((πΊ β Ring β§ (π₯ β β€ β§ π¦ β (BaseβπΊ) β§ π§ β (BaseβπΊ))) β (π¦(.rβπΊ)(π₯(.gβπΊ)π§)) = (π₯(.gβπΊ)(π¦(.rβπΊ)π§))) |
25 | 4, 5, 7, 10, 13, 16, 20, 22, 23, 24 | isassad 21286 | . 2 β’ (πΊ β Ring β π β AssAlg) |
26 | assaring 21283 | . . 3 β’ (π β AssAlg β π β Ring) | |
27 | 26, 19 | sylibr 233 | . 2 β’ (π β AssAlg β πΊ β Ring) |
28 | 25, 27 | impbii 208 | 1 β’ (πΊ β Ring β π β AssAlg) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1542 β wcel 2107 βcfv 6497 β€cz 12504 Basecbs 17088 +gcplusg 17138 .rcmulr 17139 Β·π cvsca 17142 .gcmg 18877 Abelcabl 19568 Ringcrg 19969 CRingccrg 19970 LModclmod 20336 β€ringczring 20885 β€Modczlm 20917 AssAlgcasa 21272 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-mulg 18878 df-subg 18930 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-subrg 20234 df-lmod 20338 df-cnfld 20813 df-zring 20886 df-zlm 20921 df-assa 21275 |
This theorem is referenced by: (None) |
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