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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. (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 2823 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | zlmbas 20667 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (Base‘𝐺) = (Base‘𝑊)) |
5 | eqid 2823 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | zlmplusg 20668 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝑊) |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (+g‘𝐺) = (+g‘𝑊)) |
8 | 1 | zlmsca 20670 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring = (Scalar‘𝑊)) |
9 | eqid 2823 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
10 | 1, 9 | zlmvsca 20671 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) |
12 | zringbas 20625 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤ = (Base‘ℤring)) |
14 | zringplusg 20626 | . . . 4 ⊢ + = (+g‘ℤring) | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → + = (+g‘ℤring)) |
16 | zringmulr 20628 | . . . 4 ⊢ · = (.r‘ℤring) | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → · = (.r‘ℤring)) |
18 | zring1 20630 | . . . 4 ⊢ 1 = (1r‘ℤring) | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → 1 = (1r‘ℤring)) |
20 | zringring 20622 | . . . 4 ⊢ ℤring ∈ Ring | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring ∈ Ring) |
22 | 3, 6 | ablprop 18920 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ Abel) |
23 | ablgrp 18913 | . . . 4 ⊢ (𝑊 ∈ Abel → 𝑊 ∈ Grp) | |
24 | 22, 23 | sylbi 219 | . . 3 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ Grp) |
25 | ablgrp 18913 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
26 | 2, 9 | mulgcl 18247 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
27 | 25, 26 | syl3an1 1159 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
28 | 2, 9, 5 | mulgdi 18949 | . . 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 18261 | . . . 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 582 | . . 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 18266 | . . . 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 582 | . . 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 18237 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝐺) → (1(.g‘𝐺)𝑥) = 𝑥) |
34 | 33 | adantl 484 | . . 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 19642 | . 2 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ LMod) |
36 | lmodabl 19683 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
37 | 36, 22 | sylibr 236 | . 2 ⊢ (𝑊 ∈ LMod → 𝐺 ∈ Abel) |
38 | 35, 37 | impbii 211 | 1 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 1c1 10540 + caddc 10542 · cmul 10544 ℤcz 11984 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 ·𝑠 cvsca 16571 Grpcgrp 18105 .gcmg 18226 Abelcabl 18909 1rcur 19253 Ringcrg 19299 LModclmod 19636 ℤringzring 20619 ℤModczlm 20650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-lmod 19638 df-cnfld 20548 df-zring 20620 df-zlm 20654 |
This theorem is referenced by: zlmassa 20673 zlmclm 23718 nmmulg 31211 cnzh 31213 rezh 31214 |
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