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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 2825 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | zlmbas 20226 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) = (Base‘𝑊)) |
5 | 1 | zlmsca 20229 | . . 3 ⊢ (𝐺 ∈ Ring → ℤring = (Scalar‘𝑊)) |
6 | zringbas 20184 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → ℤ = (Base‘ℤring)) |
8 | eqid 2825 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
9 | 1, 8 | zlmvsca 20230 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) |
11 | eqid 2825 | . . . . 5 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
12 | 1, 11 | zlmmulr 20228 | . . . 4 ⊢ (.r‘𝐺) = (.r‘𝑊) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → (.r‘𝐺) = (.r‘𝑊)) |
14 | ringabl 18934 | . . . 4 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Abel) | |
15 | 1 | zlmlmod 20231 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) |
16 | 14, 15 | sylib 210 | . . 3 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ LMod) |
17 | eqid 2825 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
18 | 1, 17 | zlmplusg 20227 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝑊) |
19 | 3, 18, 12 | ringprop 18938 | . . . 4 ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ Ring) |
20 | 19 | biimpi 208 | . . 3 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ Ring) |
21 | zringcrng 20180 | . . . 4 ⊢ ℤring ∈ CRing | |
22 | 21 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Ring → ℤring ∈ CRing) |
23 | 2, 8, 11 | mulgass2 18955 | . . 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 18991 | . . 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 19684 | . 2 ⊢ (𝐺 ∈ Ring → 𝑊 ∈ AssAlg) |
26 | assaring 19681 | . . 3 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
27 | 26, 19 | sylibr 226 | . 2 ⊢ (𝑊 ∈ AssAlg → 𝐺 ∈ Ring) |
28 | 25, 27 | impbii 201 | 1 ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 ℤcz 11704 Basecbs 16222 +gcplusg 16305 .rcmulr 16306 ·𝑠 cvsca 16309 .gcmg 17894 Abelcabl 18547 Ringcrg 18901 CRingccrg 18902 LModclmod 19219 AssAlgcasa 19670 ℤringzring 20178 ℤModczlm 20209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-mulg 17895 df-subg 17942 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-oppr 18977 df-subrg 19134 df-lmod 19221 df-assa 19673 df-cnfld 20107 df-zring 20179 df-zlm 20213 |
This theorem is referenced by: (None) |
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