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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 2735 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
3 | eqid 2735 | . 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁) | |
4 | eqid 2735 | . 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
5 | eqid 2735 | . 2 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁) | |
6 | eqid 2735 | . 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 33366 | . 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 7466 | . . . 4 ⊢ (𝜑 → (𝑀 ~QG 𝐺) ∈ V) | |
14 | 11, 12, 13, 7, 4 | quss 17593 | . . 3 ⊢ (𝜑 → (Scalar‘𝑀) = (Scalar‘𝑁)) |
15 | 14 | eqcomd 2741 | . 2 ⊢ (𝜑 → (Scalar‘𝑁) = (Scalar‘𝑀)) |
16 | eqid 2735 | . . . . . 6 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
17 | 16 | lsssubg 20973 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀)) |
18 | 7, 9, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀)) |
19 | lmodabl 20924 | . . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Abel) | |
20 | ablnsg 19880 | . . . . 5 ⊢ (𝑀 ∈ Abel → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) | |
21 | 7, 19, 20 | 3syl 18 | . . . 4 ⊢ (𝜑 → (NrmSGrp‘𝑀) = (SubGrp‘𝑀)) |
22 | 18, 21 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (NrmSGrp‘𝑀)) |
23 | quslmhm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
24 | 1, 8, 23 | qusghm 19286 | . . 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 17589 | . . . . 5 ⊢ (𝜑 → 𝑁 = (𝐹 “s 𝑀)) |
27 | 11, 12, 23, 13, 7 | quslem 17590 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / (𝑀 ~QG 𝐺))) |
28 | eqid 2735 | . . . . . 6 ⊢ (𝑀 ~QG 𝐺) = (𝑀 ~QG 𝐺) | |
29 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑀 ∈ LMod) |
30 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝐺 ∈ (LSubSp‘𝑀)) |
31 | simpr1 1193 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑘 ∈ (Base‘(Scalar‘𝑀))) | |
32 | simpr2 1194 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉) | |
33 | simpr3 1195 | . . . . . 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 33359 | . . . . 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 17585 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑧 ∈ 𝑉) → (𝑦( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑦( ·𝑠 ‘𝑀)𝑧))) |
36 | 35 | 3expb 1119 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑦( ·𝑠 ‘𝑀)𝑧))) |
37 | 36 | eqcomd 2741 | . 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 21056 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑀 LMHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 [cec 8742 / cqs 8743 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 /s cqus 17552 SubGrpcsubg 19151 NrmSGrpcnsg 19152 ~QG cqg 19153 GrpHom cghm 19243 Abelcabl 19814 LModclmod 20875 LSubSpclss 20947 LMHom clmhm 21036 |
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-rep 5285 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 |
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-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 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-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-lmod 20877 df-lss 20948 df-lmhm 21039 |
This theorem is referenced by: qusdimsum 33656 |
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