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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. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
zlmlmod.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
Ref | Expression |
---|---|
zlmassa | ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmlmod.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
2 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | zlmbas 20595 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) = (Base‘𝑊)) |
5 | 1 | zlmsca 20598 | . . 3 ⊢ (𝐺 ∈ Ring → ℤring = (Scalar‘𝑊)) |
6 | zringbas 20553 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → ℤ = (Base‘ℤring)) |
8 | eqid 2821 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
9 | 1, 8 | zlmvsca 20599 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) |
11 | eqid 2821 | . . . . 5 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
12 | 1, 11 | zlmmulr 20597 | . . . 4 ⊢ (.r‘𝐺) = (.r‘𝑊) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (.r‘𝐺) = (.r‘𝑊)) |
14 | ringabl 19261 | . . . 4 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Abel) | |
15 | 1 | zlmlmod 20600 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) |
16 | 14, 15 | sylib 219 | . . 3 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ LMod) |
17 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
18 | 1, 17 | zlmplusg 20596 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝑊) |
19 | 3, 18, 12 | ringprop 19265 | . . . 4 ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ Ring) |
20 | 19 | biimpi 217 | . . 3 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ Ring) |
21 | zringcrng 20549 | . . . 4 ⊢ ℤring ∈ CRing | |
22 | 21 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → ℤring ∈ CRing) |
23 | 2, 8, 11 | mulgass2 19282 | . . 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))) |
24 | 2, 8, 11 | mulgass3 19318 | . . 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))) |
25 | 4, 5, 7, 10, 13, 16, 20, 22, 23, 24 | isassad 20026 | . 2 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ AssAlg) |
26 | assaring 20023 | . . 3 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
27 | 26, 19 | sylibr 235 | . 2 ⊢ (𝑊 ∈ AssAlg → 𝐺 ∈ Ring) |
28 | 25, 27 | impbii 210 | 1 ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 ℤcz 11970 Basecbs 16473 +gcplusg 16555 .rcmulr 16556 ·𝑠 cvsca 16559 .gcmg 18164 Abelcabl 18838 Ringcrg 19228 CRingccrg 19229 LModclmod 19565 AssAlgcasa 20012 ℤringzring 20547 ℤModczlm 20578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-tpos 7883 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-fz 12883 df-fzo 13024 df-seq 13360 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-0g 16705 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-grp 18046 df-minusg 18047 df-mulg 18165 df-subg 18216 df-cmn 18839 df-abl 18840 df-mgp 19171 df-ur 19183 df-ring 19230 df-cring 19231 df-oppr 19304 df-subrg 19464 df-lmod 19567 df-assa 20015 df-cnfld 20476 df-zring 20548 df-zlm 20582 |
This theorem is referenced by: (None) |
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