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Mirrors > Home > MPE Home > Th. List > zlmlmod | Structured version Visualization version GIF version |
Description: The ℤ-module operation turns an arbitrary abelian group into a left module over ℤ. Also see zlmassa 20588. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
zlmlmod.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
Ref | Expression |
---|---|
zlmlmod | ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmlmod.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
2 | eqid 2798 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | zlmbas 20211 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (Base‘𝐺) = (Base‘𝑊)) |
5 | eqid 2798 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | zlmplusg 20212 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝑊) |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (+g‘𝐺) = (+g‘𝑊)) |
8 | 1 | zlmsca 20214 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring = (Scalar‘𝑊)) |
9 | eqid 2798 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
10 | 1, 9 | zlmvsca 20215 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) |
12 | zringbas 20169 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤ = (Base‘ℤring)) |
14 | zringplusg 20170 | . . . 4 ⊢ + = (+g‘ℤring) | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → + = (+g‘ℤring)) |
16 | zringmulr 20172 | . . . 4 ⊢ · = (.r‘ℤring) | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → · = (.r‘ℤring)) |
18 | zring1 20174 | . . . 4 ⊢ 1 = (1r‘ℤring) | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → 1 = (1r‘ℤring)) |
20 | zringring 20166 | . . . 4 ⊢ ℤring ∈ Ring | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring ∈ Ring) |
22 | 3, 6 | ablprop 18910 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ Abel) |
23 | ablgrp 18903 | . . . 4 ⊢ (𝑊 ∈ Abel → 𝑊 ∈ Grp) | |
24 | 22, 23 | sylbi 220 | . . 3 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ Grp) |
25 | ablgrp 18903 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
26 | 2, 9 | mulgcl 18237 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
27 | 25, 26 | syl3an1 1160 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
28 | 2, 9, 5 | mulgdi 18940 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥(.g‘𝐺)(𝑦(+g‘𝐺)𝑧)) = ((𝑥(.g‘𝐺)𝑦)(+g‘𝐺)(𝑥(.g‘𝐺)𝑧))) |
29 | 2, 9, 5 | mulgdir 18251 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥 + 𝑦)(.g‘𝐺)𝑧) = ((𝑥(.g‘𝐺)𝑧)(+g‘𝐺)(𝑦(.g‘𝐺)𝑧))) |
30 | 25, 29 | sylan 583 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥 + 𝑦)(.g‘𝐺)𝑧) = ((𝑥(.g‘𝐺)𝑧)(+g‘𝐺)(𝑦(.g‘𝐺)𝑧))) |
31 | 2, 9 | mulgass 18256 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥 · 𝑦)(.g‘𝐺)𝑧) = (𝑥(.g‘𝐺)(𝑦(.g‘𝐺)𝑧))) |
32 | 25, 31 | sylan 583 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥 · 𝑦)(.g‘𝐺)𝑧) = (𝑥(.g‘𝐺)(𝑦(.g‘𝐺)𝑧))) |
33 | 2, 9 | mulg1 18227 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝐺) → (1(.g‘𝐺)𝑥) = 𝑥) |
34 | 33 | adantl 485 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺)) → (1(.g‘𝐺)𝑥) = 𝑥) |
35 | 4, 7, 8, 11, 13, 15, 17, 19, 21, 24, 27, 28, 30, 32, 34 | islmodd 19633 | . 2 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ LMod) |
36 | lmodabl 19674 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
37 | 36, 22 | sylibr 237 | . 2 ⊢ (𝑊 ∈ LMod → 𝐺 ∈ Abel) |
38 | 35, 37 | impbii 212 | 1 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 · cmul 10531 ℤcz 11969 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 ·𝑠 cvsca 16561 Grpcgrp 18095 .gcmg 18216 Abelcabl 18899 1rcur 19244 Ringcrg 19290 LModclmod 19627 ℤringzring 20163 ℤModczlm 20194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-lmod 19629 df-cnfld 20092 df-zring 20164 df-zlm 20198 |
This theorem is referenced by: zlmassa 20588 zlmclm 23717 nmmulg 31319 cnzh 31321 rezh 31322 |
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