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Mirrors > Home > MPE Home > Th. List > invlmhm | Structured version Visualization version GIF version |
Description: The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
invlmhm.b | β’ πΌ = (invgβπ) |
Ref | Expression |
---|---|
invlmhm | β’ (π β LMod β πΌ β (π LMHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2733 | . 2 β’ ( Β·π βπ) = ( Β·π βπ) | |
3 | eqid 2733 | . 2 β’ (Scalarβπ) = (Scalarβπ) | |
4 | eqid 2733 | . 2 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
5 | id 22 | . 2 β’ (π β LMod β π β LMod) | |
6 | eqidd 2734 | . 2 β’ (π β LMod β (Scalarβπ) = (Scalarβπ)) | |
7 | lmodabl 20384 | . . 3 β’ (π β LMod β π β Abel) | |
8 | invlmhm.b | . . . 4 β’ πΌ = (invgβπ) | |
9 | 1, 8 | invghm 19617 | . . 3 β’ (π β Abel β πΌ β (π GrpHom π)) |
10 | 7, 9 | sylib 217 | . 2 β’ (π β LMod β πΌ β (π GrpHom π)) |
11 | 1, 3, 2, 8, 4 | lmodvsinv2 20513 | . . . 4 β’ ((π β LMod β§ π₯ β (Baseβ(Scalarβπ)) β§ π¦ β (Baseβπ)) β (π₯( Β·π βπ)(πΌβπ¦)) = (πΌβ(π₯( Β·π βπ)π¦))) |
12 | 11 | eqcomd 2739 | . . 3 β’ ((π β LMod β§ π₯ β (Baseβ(Scalarβπ)) β§ π¦ β (Baseβπ)) β (πΌβ(π₯( Β·π βπ)π¦)) = (π₯( Β·π βπ)(πΌβπ¦))) |
13 | 12 | 3expb 1121 | . 2 β’ ((π β LMod β§ (π₯ β (Baseβ(Scalarβπ)) β§ π¦ β (Baseβπ))) β (πΌβ(π₯( Β·π βπ)π¦)) = (π₯( Β·π βπ)(πΌβπ¦))) |
14 | 1, 2, 2, 3, 3, 4, 5, 5, 6, 10, 13 | islmhmd 20515 | 1 β’ (π β LMod β πΌ β (π LMHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 invgcminusg 18754 GrpHom cghm 19010 Abelcabl 19568 LModclmod 20336 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-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-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 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-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-ghm 19011 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lmhm 20498 |
This theorem is referenced by: mendring 41562 |
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