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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 2824 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
6 | ovexd 7194 | . . 3 ⊢ (𝜑 → (𝑀 ~QG 𝐺) ∈ V) | |
7 | quslmod.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
8 | 2, 4, 5, 6, 7 | qusval 16818 | . 2 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) “s 𝑀)) |
9 | eqid 2824 | . 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
10 | eqid 2824 | . 2 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
11 | eqid 2824 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
12 | eqid 2824 | . 2 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
13 | 2, 4, 5, 6, 7 | quslem 16819 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)):𝑉–onto→(𝑉 / (𝑀 ~QG 𝐺))) |
14 | quslmod.2 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) | |
15 | eqid 2824 | . . . . . 6 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
16 | 15 | lsssubg 19732 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀)) |
17 | 7, 14, 16 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀)) |
18 | eqid 2824 | . . . . 5 ⊢ (𝑀 ~QG 𝐺) = (𝑀 ~QG 𝐺) | |
19 | 3, 18 | eqger 18333 | . . . 4 ⊢ (𝐺 ∈ (SubGrp‘𝑀) → (𝑀 ~QG 𝐺) Er 𝑉) |
20 | 17, 19 | syl 17 | . . 3 ⊢ (𝜑 → (𝑀 ~QG 𝐺) Er 𝑉) |
21 | 3 | fvexi 6687 | . . . 4 ⊢ 𝑉 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
23 | lmodgrp 19644 | . . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
24 | 7, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Grp) |
25 | 24 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 𝑀 ∈ Grp) |
26 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 𝑝 ∈ 𝑉) | |
27 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 𝑞 ∈ 𝑉) | |
28 | 3, 10 | grpcl 18114 | . . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉) → (𝑝(+g‘𝑀)𝑞) ∈ 𝑉) |
29 | 25, 26, 27, 28 | syl3anc 1367 | . . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝(+g‘𝑀)𝑞) ∈ 𝑉) |
30 | lmodabl 19684 | . . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Abel) | |
31 | ablnsg 18970 | . . . . . 6 ⊢ (𝑀 ∈ Abel → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) | |
32 | 7, 30, 31 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) |
33 | 17, 32 | eleqtrrd 2919 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (NrmSGrp‘𝑀)) |
34 | 3, 18, 10 | eqgcpbl 18337 | . . . 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 16825 | . 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 483 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑀 ∈ LMod) |
38 | 14 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐺 ∈ (LSubSp‘𝑀)) |
39 | simpr1 1190 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑘 ∈ (Base‘(Scalar‘𝑀))) | |
40 | eqid 2824 | . . 3 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁) | |
41 | simpr2 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉) | |
42 | simpr3 1192 | . . 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 30924 | . 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 30926 | 1 ⊢ (𝜑 → 𝑁 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 Er wer 8289 [cec 8290 / cqs 8291 Basecbs 16486 +gcplusg 16568 Scalarcsca 16571 ·𝑠 cvsca 16572 0gc0g 16716 /s cqus 16781 Grpcgrp 18106 SubGrpcsubg 18276 NrmSGrpcnsg 18277 ~QG cqg 18278 Abelcabl 18910 LModclmod 19637 LSubSpclss 19706 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-ec 8294 df-qs 8298 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-0g 16718 df-imas 16784 df-qus 16785 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-nsg 18280 df-eqg 18281 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 df-lss 19707 |
This theorem is referenced by: quslmhm 30928 |
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