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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 21836. (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 2728 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | zlmbas 21444 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (Base‘𝐺) = (Base‘𝑊)) |
5 | eqid 2728 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | zlmplusg 21446 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝑊) |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (+g‘𝐺) = (+g‘𝑊)) |
8 | 1 | zlmsca 21450 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring = (Scalar‘𝑊)) |
9 | eqid 2728 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
10 | 1, 9 | zlmvsca 21451 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) |
12 | zringbas 21379 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤ = (Base‘ℤring)) |
14 | zringplusg 21380 | . . . 4 ⊢ + = (+g‘ℤring) | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → + = (+g‘ℤring)) |
16 | zringmulr 21383 | . . . 4 ⊢ · = (.r‘ℤring) | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → · = (.r‘ℤring)) |
18 | zring1 21385 | . . . 4 ⊢ 1 = (1r‘ℤring) | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → 1 = (1r‘ℤring)) |
20 | zringring 21375 | . . . 4 ⊢ ℤring ∈ Ring | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring ∈ Ring) |
22 | 3, 6 | ablprop 19748 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ Abel) |
23 | ablgrp 19740 | . . . 4 ⊢ (𝑊 ∈ Abel → 𝑊 ∈ Grp) | |
24 | 22, 23 | sylbi 216 | . . 3 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ Grp) |
25 | ablgrp 19740 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
26 | 2, 9 | mulgcl 19046 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
27 | 25, 26 | syl3an1 1161 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
28 | 2, 9, 5 | mulgdi 19781 | . . 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 19061 | . . . 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 579 | . . 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 19066 | . . . 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 579 | . . 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 19036 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝐺) → (1(.g‘𝐺)𝑥) = 𝑥) |
34 | 33 | adantl 481 | . . 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 20749 | . 2 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ LMod) |
36 | lmodabl 20792 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
37 | 36, 22 | sylibr 233 | . 2 ⊢ (𝑊 ∈ LMod → 𝐺 ∈ Abel) |
38 | 35, 37 | impbii 208 | 1 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 1c1 11140 + caddc 11142 · cmul 11144 ℤcz 12589 Basecbs 17180 +gcplusg 17233 .rcmulr 17234 ·𝑠 cvsca 17237 Grpcgrp 18890 .gcmg 19023 Abelcabl 19736 1rcur 20121 Ringcrg 20173 LModclmod 20743 ℤringczring 21372 ℤModczlm 21426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-cnfld 21280 df-zring 21373 df-zlm 21430 |
This theorem is referenced by: zlmassa 21836 zlmclm 25052 nmmulg 33569 cnzh 33571 rezh 33572 |
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