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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 21923. (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 2737 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 3 | 1, 2 | zlmbas 21529 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑊) |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (Base‘𝐺) = (Base‘𝑊)) |
| 5 | eqid 2737 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 6 | 1, 5 | zlmplusg 21531 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝑊) |
| 7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (+g‘𝐺) = (+g‘𝑊)) |
| 8 | 1 | zlmsca 21535 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring = (Scalar‘𝑊)) |
| 9 | eqid 2737 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 10 | 1, 9 | zlmvsca 21536 | . . . 4 ⊢ (.g‘𝐺) = ( ·𝑠 ‘𝑊) |
| 11 | 10 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → (.g‘𝐺) = ( ·𝑠 ‘𝑊)) |
| 12 | zringbas 21464 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
| 13 | 12 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤ = (Base‘ℤring)) |
| 14 | zringplusg 21465 | . . . 4 ⊢ + = (+g‘ℤring) | |
| 15 | 14 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → + = (+g‘ℤring)) |
| 16 | zringmulr 21468 | . . . 4 ⊢ · = (.r‘ℤring) | |
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → · = (.r‘ℤring)) |
| 18 | zring1 21470 | . . . 4 ⊢ 1 = (1r‘ℤring) | |
| 19 | 18 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → 1 = (1r‘ℤring)) |
| 20 | zringring 21460 | . . . 4 ⊢ ℤring ∈ Ring | |
| 21 | 20 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Abel → ℤring ∈ Ring) |
| 22 | 3, 6 | ablprop 19811 | . . . 4 ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ Abel) |
| 23 | ablgrp 19803 | . . . 4 ⊢ (𝑊 ∈ Abel → 𝑊 ∈ Grp) | |
| 24 | 22, 23 | sylbi 217 | . . 3 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ Grp) |
| 25 | ablgrp 19803 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 26 | 2, 9 | mulgcl 19109 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
| 27 | 25, 26 | syl3an1 1164 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(.g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
| 28 | 2, 9, 5 | mulgdi 19844 | . . 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 19124 | . . . 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 19129 | . . . 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 19099 | . . . 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 20864 | . 2 ⊢ (𝐺 ∈ Abel → 𝑊 ∈ LMod) |
| 36 | lmodabl 20907 | . . 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 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 · cmul 11160 ℤcz 12613 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 ·𝑠 cvsca 17301 Grpcgrp 18951 .gcmg 19085 Abelcabl 19799 1rcur 20178 Ringcrg 20230 LModclmod 20858 ℤringczring 21457 ℤModczlm 21511 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-cnfld 21365 df-zring 21458 df-zlm 21515 |
| This theorem is referenced by: zlmassa 21923 zlmclm 25145 nmmulg 33967 cnzh 33969 rezh 33970 |
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