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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 21941. (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 2735 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | zlmbas 21547 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (Base‘𝐺) = (Base‘𝑊)) |
5 | eqid 2735 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | zlmplusg 21549 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝑊) |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (+g‘𝐺) = (+g‘𝑊)) |
8 | 1 | zlmsca 21553 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring = (Scalar‘𝑊)) |
9 | eqid 2735 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
10 | 1, 9 | zlmvsca 21554 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) |
12 | zringbas 21482 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤ = (Base‘ℤring)) |
14 | zringplusg 21483 | . . . 4 ⊢ + = (+g‘ℤring) | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → + = (+g‘ℤring)) |
16 | zringmulr 21486 | . . . 4 ⊢ · = (.r‘ℤring) | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → · = (.r‘ℤring)) |
18 | zring1 21488 | . . . 4 ⊢ 1 = (1r‘ℤring) | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → 1 = (1r‘ℤring)) |
20 | zringring 21478 | . . . 4 ⊢ ℤring ∈ Ring | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring ∈ Ring) |
22 | 3, 6 | ablprop 19826 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ Abel) |
23 | ablgrp 19818 | . . . 4 ⊢ (𝑊 ∈ Abel → 𝑊 ∈ Grp) | |
24 | 22, 23 | sylbi 217 | . . 3 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ Grp) |
25 | ablgrp 19818 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
26 | 2, 9 | mulgcl 19122 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
27 | 25, 26 | syl3an1 1162 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
28 | 2, 9, 5 | mulgdi 19859 | . . 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 19137 | . . . 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 580 | . . 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 19142 | . . . 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 580 | . . 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 19112 | . . . 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 20881 | . 2 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ LMod) |
36 | lmodabl 20924 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
37 | 36, 22 | sylibr 234 | . 2 ⊢ (𝑊 ∈ LMod → 𝐺 ∈ Abel) |
38 | 35, 37 | impbii 209 | 1 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 · cmul 11158 ℤcz 12611 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 ·𝑠 cvsca 17302 Grpcgrp 18964 .gcmg 19098 Abelcabl 19814 1rcur 20199 Ringcrg 20251 LModclmod 20875 ℤringczring 21475 ℤModczlm 21529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-cnfld 21383 df-zring 21476 df-zlm 21533 |
This theorem is referenced by: zlmassa 21941 zlmclm 25159 nmmulg 33929 cnzh 33931 rezh 33932 |
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