Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > quslmhm | Structured version Visualization version GIF version |
Description: If 𝐺 is a submodule of 𝑀, then the "natural map" from elements to their cosets is a left module homomorphism from 𝑀 to 𝑀 / 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)) |
quslmhm.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) |
Ref | Expression |
---|---|
quslmhm | ⊢ (𝜑 → 𝐹 ∈ (𝑀 LMHom 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | quslmod.v | . 2 ⊢ 𝑉 = (Base‘𝑀) | |
2 | eqid 2823 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
3 | eqid 2823 | . 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁) | |
4 | eqid 2823 | . 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
5 | eqid 2823 | . 2 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁) | |
6 | eqid 2823 | . 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
7 | quslmod.1 | . 2 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
8 | quslmod.n | . . 3 ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) | |
9 | quslmod.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) | |
10 | 8, 1, 7, 9 | quslmod 30925 | . 2 ⊢ (𝜑 → 𝑁 ∈ LMod) |
11 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))) |
12 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑀)) |
13 | ovexd 7193 | . . . 4 ⊢ (𝜑 → (𝑀 ~QG 𝐺) ∈ V) | |
14 | 11, 12, 13, 7, 4 | quss 16821 | . . 3 ⊢ (𝜑 → (Scalar‘𝑀) = (Scalar‘𝑁)) |
15 | 14 | eqcomd 2829 | . 2 ⊢ (𝜑 → (Scalar‘𝑁) = (Scalar‘𝑀)) |
16 | eqid 2823 | . . . . . 6 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
17 | 16 | lsssubg 19731 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀)) |
18 | 7, 9, 17 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀)) |
19 | lmodabl 19683 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Abel) | |
20 | ablnsg 18969 | . . . . 5 ⊢ (𝑀 ∈ Abel → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) | |
21 | 7, 19, 20 | 3syl 18 | . . . 4 ⊢ (𝜑 → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) |
22 | 18, 21 | eleqtrrd 2918 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (NrmSGrp‘𝑀)) |
23 | quslmhm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
24 | 1, 8, 23 | qusghm 18397 | . . 3 ⊢ (𝐺 ∈ (NrmSGrp‘𝑀) → 𝐹 ∈ (𝑀 GrpHom 𝑁)) |
25 | 22, 24 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑀 GrpHom 𝑁)) |
26 | 11, 12, 23, 13, 7 | qusval 16817 | . . . . 5 ⊢ (𝜑 → 𝑁 = (𝐹 “s 𝑀)) |
27 | 11, 12, 23, 13, 7 | quslem 16818 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / (𝑀 ~QG 𝐺))) |
28 | eqid 2823 | . . . . . 6 ⊢ (𝑀 ~QG 𝐺) = (𝑀 ~QG 𝐺) | |
29 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑀 ∈ LMod) |
30 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝐺 ∈ (LSubSp‘𝑀)) |
31 | simpr1 1190 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑘 ∈ (Base‘(Scalar‘𝑀))) | |
32 | simpr2 1191 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉) | |
33 | simpr3 1192 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉) | |
34 | 1, 28, 6, 2, 29, 30, 31, 8, 3, 23, 32, 33 | qusvscpbl 30922 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑢) = (𝐹‘𝑣) → (𝐹‘(𝑘( ·𝑠 ‘𝑀)𝑢)) = (𝐹‘(𝑘( ·𝑠 ‘𝑀)𝑣)))) |
35 | 26, 12, 27, 7, 4, 6, 2, 3, 34 | imasvscaval 16813 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑧 ∈ 𝑉) → (𝑦( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑦( ·𝑠 ‘𝑀)𝑧))) |
36 | 35 | 3expb 1116 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑦( ·𝑠 ‘𝑀)𝑧))) |
37 | 36 | eqcomd 2829 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑦( ·𝑠 ‘𝑀)𝑧)) = (𝑦( ·𝑠 ‘𝑁)(𝐹‘𝑧))) |
38 | 1, 2, 3, 4, 5, 6, 7, 10, 15, 25, 37 | islmhmd 19813 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑀 LMHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 [cec 8289 / cqs 8290 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 /s cqus 16780 SubGrpcsubg 18275 NrmSGrpcnsg 18276 ~QG cqg 18277 GrpHom cghm 18357 Abelcabl 18909 LModclmod 19636 LSubSpclss 19705 LMHom clmhm 19793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-lmod 19638 df-lss 19706 df-lmhm 19796 |
This theorem is referenced by: qusdimsum 31026 |
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