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Mirrors > Home > MPE Home > Th. List > Mathboxes > quslmhm | Structured version Visualization version GIF version |
Description: If πΊ is a submodule of π, then the "natural map" from elements to their cosets is a left module homomorphism from π to π / πΊ. (Contributed by Thierry Arnoux, 18-May-2023.) |
Ref | Expression |
---|---|
quslmod.n | β’ π = (π /s (π ~QG πΊ)) |
quslmod.v | β’ π = (Baseβπ) |
quslmod.1 | β’ (π β π β LMod) |
quslmod.2 | β’ (π β πΊ β (LSubSpβπ)) |
quslmhm.f | β’ πΉ = (π₯ β π β¦ [π₯](π ~QG πΊ)) |
Ref | Expression |
---|---|
quslmhm | β’ (π β πΉ β (π LMHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | quslmod.v | . 2 β’ π = (Baseβπ) | |
2 | eqid 2733 | . 2 β’ ( Β·π βπ) = ( Β·π βπ) | |
3 | eqid 2733 | . 2 β’ ( Β·π βπ) = ( Β·π βπ) | |
4 | eqid 2733 | . 2 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2733 | . 2 β’ (Scalarβπ) = (Scalarβπ) | |
6 | eqid 2733 | . 2 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
7 | quslmod.1 | . 2 β’ (π β π β LMod) | |
8 | quslmod.n | . . 3 β’ π = (π /s (π ~QG πΊ)) | |
9 | quslmod.2 | . . 3 β’ (π β πΊ β (LSubSpβπ)) | |
10 | 8, 1, 7, 9 | quslmod 32193 | . 2 β’ (π β π β LMod) |
11 | 8 | a1i 11 | . . . 4 β’ (π β π = (π /s (π ~QG πΊ))) |
12 | 1 | a1i 11 | . . . 4 β’ (π β π = (Baseβπ)) |
13 | ovexd 7393 | . . . 4 β’ (π β (π ~QG πΊ) β V) | |
14 | 11, 12, 13, 7, 4 | quss 17433 | . . 3 β’ (π β (Scalarβπ) = (Scalarβπ)) |
15 | 14 | eqcomd 2739 | . 2 β’ (π β (Scalarβπ) = (Scalarβπ)) |
16 | eqid 2733 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
17 | 16 | lsssubg 20433 | . . . . 5 β’ ((π β LMod β§ πΊ β (LSubSpβπ)) β πΊ β (SubGrpβπ)) |
18 | 7, 9, 17 | syl2anc 585 | . . . 4 β’ (π β πΊ β (SubGrpβπ)) |
19 | lmodabl 20384 | . . . . 5 β’ (π β LMod β π β Abel) | |
20 | ablnsg 19630 | . . . . 5 β’ (π β Abel β (NrmSGrpβπ) = (SubGrpβπ)) | |
21 | 7, 19, 20 | 3syl 18 | . . . 4 β’ (π β (NrmSGrpβπ) = (SubGrpβπ)) |
22 | 18, 21 | eleqtrrd 2837 | . . 3 β’ (π β πΊ β (NrmSGrpβπ)) |
23 | quslmhm.f | . . . 4 β’ πΉ = (π₯ β π β¦ [π₯](π ~QG πΊ)) | |
24 | 1, 8, 23 | qusghm 19050 | . . 3 β’ (πΊ β (NrmSGrpβπ) β πΉ β (π GrpHom π)) |
25 | 22, 24 | syl 17 | . 2 β’ (π β πΉ β (π GrpHom π)) |
26 | 11, 12, 23, 13, 7 | qusval 17429 | . . . . 5 β’ (π β π = (πΉ βs π)) |
27 | 11, 12, 23, 13, 7 | quslem 17430 | . . . . 5 β’ (π β πΉ:πβontoβ(π / (π ~QG πΊ))) |
28 | eqid 2733 | . . . . . 6 β’ (π ~QG πΊ) = (π ~QG πΊ) | |
29 | 7 | adantr 482 | . . . . . 6 β’ ((π β§ (π β (Baseβ(Scalarβπ)) β§ π’ β π β§ π£ β π)) β π β LMod) |
30 | 9 | adantr 482 | . . . . . 6 β’ ((π β§ (π β (Baseβ(Scalarβπ)) β§ π’ β π β§ π£ β π)) β πΊ β (LSubSpβπ)) |
31 | simpr1 1195 | . . . . . 6 β’ ((π β§ (π β (Baseβ(Scalarβπ)) β§ π’ β π β§ π£ β π)) β π β (Baseβ(Scalarβπ))) | |
32 | simpr2 1196 | . . . . . 6 β’ ((π β§ (π β (Baseβ(Scalarβπ)) β§ π’ β π β§ π£ β π)) β π’ β π) | |
33 | simpr3 1197 | . . . . . 6 β’ ((π β§ (π β (Baseβ(Scalarβπ)) β§ π’ β π β§ π£ β π)) β π£ β π) | |
34 | 1, 28, 6, 2, 29, 30, 31, 8, 3, 23, 32, 33 | qusvscpbl 32190 | . . . . 5 β’ ((π β§ (π β (Baseβ(Scalarβπ)) β§ π’ β π β§ π£ β π)) β ((πΉβπ’) = (πΉβπ£) β (πΉβ(π( Β·π βπ)π’)) = (πΉβ(π( Β·π βπ)π£)))) |
35 | 26, 12, 27, 7, 4, 6, 2, 3, 34 | imasvscaval 17425 | . . . 4 β’ ((π β§ π¦ β (Baseβ(Scalarβπ)) β§ π§ β π) β (π¦( Β·π βπ)(πΉβπ§)) = (πΉβ(π¦( Β·π βπ)π§))) |
36 | 35 | 3expb 1121 | . . 3 β’ ((π β§ (π¦ β (Baseβ(Scalarβπ)) β§ π§ β π)) β (π¦( Β·π βπ)(πΉβπ§)) = (πΉβ(π¦( Β·π βπ)π§))) |
37 | 36 | eqcomd 2739 | . 2 β’ ((π β§ (π¦ β (Baseβ(Scalarβπ)) β§ π§ β π)) β (πΉβ(π¦( Β·π βπ)π§)) = (π¦( Β·π βπ)(πΉβπ§))) |
38 | 1, 2, 3, 4, 5, 6, 7, 10, 15, 25, 37 | islmhmd 20515 | 1 β’ (π β πΉ β (π LMHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3444 β¦ cmpt 5189 βcfv 6497 (class class class)co 7358 [cec 8649 / cqs 8650 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 /s cqus 17392 SubGrpcsubg 18927 NrmSGrpcnsg 18928 ~QG cqg 18929 GrpHom cghm 19010 Abelcabl 19568 LModclmod 20336 LSubSpclss 20407 LMHom clmhm 20495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-ec 8653 df-qs 8657 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-0g 17328 df-imas 17395 df-qus 17396 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-nsg 18931 df-eqg 18932 df-ghm 19011 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lmhm 20498 |
This theorem is referenced by: qusdimsum 32380 |
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