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| Mirrors > Home > MPE Home > Th. List > Mathboxes > quslmod | Structured version Visualization version GIF version | ||
| Description: If 𝐺 is a submodule in 𝑀, then 𝑁 = 𝑀 / 𝐺 is a left module, called the quotient module of 𝑀 by 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.) |
| Ref | Expression |
|---|---|
| quslmod.n | ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) |
| quslmod.v | ⊢ 𝑉 = (Base‘𝑀) |
| quslmod.1 | ⊢ (𝜑 → 𝑀 ∈ LMod) |
| quslmod.2 | ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) |
| Ref | Expression |
|---|---|
| quslmod | ⊢ (𝜑 → 𝑁 ∈ LMod) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | quslmod.n | . . . 4 ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))) |
| 3 | quslmod.v | . . . 4 ⊢ 𝑉 = (Base‘𝑀) | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑀)) |
| 5 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
| 6 | ovexd 7466 | . . 3 ⊢ (𝜑 → (𝑀 ~QG 𝐺) ∈ V) | |
| 7 | quslmod.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
| 8 | 2, 4, 5, 6, 7 | qusval 17587 | . 2 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) “s 𝑀)) |
| 9 | eqid 2737 | . 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
| 10 | eqid 2737 | . 2 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 11 | eqid 2737 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
| 12 | eqid 2737 | . 2 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
| 13 | 2, 4, 5, 6, 7 | quslem 17588 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)):𝑉–onto→(𝑉 / (𝑀 ~QG 𝐺))) |
| 14 | quslmod.2 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) | |
| 15 | eqid 2737 | . . . . . 6 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
| 16 | 15 | lsssubg 20955 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀)) |
| 17 | 7, 14, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀)) |
| 18 | eqid 2737 | . . . . 5 ⊢ (𝑀 ~QG 𝐺) = (𝑀 ~QG 𝐺) | |
| 19 | 3, 18 | eqger 19196 | . . . 4 ⊢ (𝐺 ∈ (SubGrp‘𝑀) → (𝑀 ~QG 𝐺) Er 𝑉) |
| 20 | 17, 19 | syl 17 | . . 3 ⊢ (𝜑 → (𝑀 ~QG 𝐺) Er 𝑉) |
| 21 | 3 | fvexi 6920 | . . . 4 ⊢ 𝑉 ∈ V |
| 22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
| 23 | lmodgrp 20865 | . . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
| 24 | 7, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Grp) |
| 25 | 24 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 𝑀 ∈ Grp) |
| 26 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 𝑝 ∈ 𝑉) | |
| 27 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 𝑞 ∈ 𝑉) | |
| 28 | 3, 10 | grpcl 18959 | . . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉) → (𝑝(+g‘𝑀)𝑞) ∈ 𝑉) |
| 29 | 25, 26, 27, 28 | syl3anc 1373 | . . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝(+g‘𝑀)𝑞) ∈ 𝑉) |
| 30 | lmodabl 20907 | . . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Abel) | |
| 31 | ablnsg 19865 | . . . . . 6 ⊢ (𝑀 ∈ Abel → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) | |
| 32 | 7, 30, 31 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) |
| 33 | 17, 32 | eleqtrrd 2844 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (NrmSGrp‘𝑀)) |
| 34 | 3, 18, 10 | eqgcpbl 19200 | . . . 4 ⊢ (𝐺 ∈ (NrmSGrp‘𝑀) → ((𝑎(𝑀 ~QG 𝐺)𝑝 ∧ 𝑏(𝑀 ~QG 𝐺)𝑞) → (𝑎(+g‘𝑀)𝑏)(𝑀 ~QG 𝐺)(𝑝(+g‘𝑀)𝑞))) |
| 35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑎(𝑀 ~QG 𝐺)𝑝 ∧ 𝑏(𝑀 ~QG 𝐺)𝑞) → (𝑎(+g‘𝑀)𝑏)(𝑀 ~QG 𝐺)(𝑝(+g‘𝑀)𝑞))) |
| 36 | 20, 22, 5, 29, 35 | ercpbl 17594 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑎) = ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑝) ∧ ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑏) = ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑞)) → ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝑎(+g‘𝑀)𝑏)) = ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝑝(+g‘𝑀)𝑞)))) |
| 37 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑀 ∈ LMod) |
| 38 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐺 ∈ (LSubSp‘𝑀)) |
| 39 | simpr1 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑘 ∈ (Base‘(Scalar‘𝑀))) | |
| 40 | eqid 2737 | . . 3 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁) | |
| 41 | simpr2 1196 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉) | |
| 42 | simpr3 1197 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
| 43 | 3, 18, 9, 11, 37, 38, 39, 1, 40, 5, 41, 42 | qusvscpbl 33379 | . 2 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑎) = ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑏) → ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝑘( ·𝑠 ‘𝑀)𝑎)) = ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝑘( ·𝑠 ‘𝑀)𝑏)))) |
| 44 | 8, 3, 9, 10, 11, 12, 13, 36, 43, 7 | imaslmod 33381 | 1 ⊢ (𝜑 → 𝑁 ∈ LMod) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Er wer 8742 [cec 8743 / cqs 8744 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 /s cqus 17550 Grpcgrp 18951 SubGrpcsubg 19138 NrmSGrpcnsg 19139 ~QG cqg 19140 Abelcabl 19799 LModclmod 20858 LSubSpclss 20929 |
| 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-rep 5279 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 |
| 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-ec 8747 df-qs 8751 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 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-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-lmod 20860 df-lss 20930 |
| This theorem is referenced by: quslmhm 33387 quslvec 33388 lmhmqusker 33445 |
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