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| Mirrors > Home > MPE Home > Th. List > zlmassa | Structured version Visualization version GIF version | ||
| Description: The ℤ-module operation turns a ring into an associative algebra over ℤ. Also see zlmlmod 21537. (Contributed by Mario Carneiro, 2-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| zlmassa.w | ⊢ 𝑊 = (ℤMod‘𝐺) | 
| Ref | Expression | 
|---|---|
| zlmassa | ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | zlmassa.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
| 2 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 3 | 1, 2 | zlmbas 21529 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) | 
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) = (Base‘𝑊)) | 
| 5 | 1 | zlmsca 21535 | . . 3 ⊢ (𝐺 ∈ Ring → ℤring = (Scalar‘𝑊)) | 
| 6 | zringbas 21464 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
| 7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → ℤ = (Base‘ℤring)) | 
| 8 | eqid 2737 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 9 | 1, 8 | zlmvsca 21536 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) | 
| 10 | 9 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) | 
| 11 | eqid 2737 | . . . . 5 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
| 12 | 1, 11 | zlmmulr 21533 | . . . 4 ⊢ (.r‘𝐺) = (.r‘𝑊) | 
| 13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (.r‘𝐺) = (.r‘𝑊)) | 
| 14 | ringabl 20278 | . . . 4 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Abel) | |
| 15 | 1 | zlmlmod 21537 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) | 
| 16 | 14, 15 | sylib 218 | . . 3 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ LMod) | 
| 17 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 18 | 1, 17 | zlmplusg 21531 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝑊) | 
| 19 | 3, 18, 12 | ringprop 20287 | . . . 4 ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ Ring) | 
| 20 | 19 | biimpi 216 | . . 3 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ Ring) | 
| 21 | 2, 8, 11 | mulgass2 20306 | . . 3 ⊢ ((𝐺 ∈ Ring ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥(.g‘𝐺)𝑦)(.r‘𝐺)𝑧) = (𝑥(.g‘𝐺)(𝑦(.r‘𝐺)𝑧))) | 
| 22 | 2, 8, 11 | mulgass3 20353 | . . 3 ⊢ ((𝐺 ∈ Ring ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(.r‘𝐺)(𝑥(.g‘𝐺)𝑧)) = (𝑥(.g‘𝐺)(𝑦(.r‘𝐺)𝑧))) | 
| 23 | 4, 5, 7, 10, 13, 16, 20, 21, 22 | isassad 21885 | . 2 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ AssAlg) | 
| 24 | assaring 21881 | . . 3 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
| 25 | 24, 19 | sylibr 234 | . 2 ⊢ (𝑊 ∈ AssAlg → 𝐺 ∈ Ring) | 
| 26 | 23, 25 | impbii 209 | 1 ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 ℤcz 12613 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 ·𝑠 cvsca 17301 .gcmg 19085 Abelcabl 19799 Ringcrg 20230 LModclmod 20858 ℤringczring 21457 ℤModczlm 21511 AssAlgcasa 21870 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-cnfld 21365 df-zring 21458 df-zlm 21515 df-assa 21873 | 
| This theorem is referenced by: (None) | 
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