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Mirrors > Home > MPE Home > Th. List > lvecvscan2 | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan2 30326 analog.) (Contributed by NM, 2-Jul-2014.) |
Ref | Expression |
---|---|
lvecmulcan2.v | β’ π = (Baseβπ) |
lvecmulcan2.s | β’ Β· = ( Β·π βπ) |
lvecmulcan2.f | β’ πΉ = (Scalarβπ) |
lvecmulcan2.k | β’ πΎ = (BaseβπΉ) |
lvecmulcan2.o | β’ 0 = (0gβπ) |
lvecmulcan2.w | β’ (π β π β LVec) |
lvecmulcan2.a | β’ (π β π΄ β πΎ) |
lvecmulcan2.b | β’ (π β π΅ β πΎ) |
lvecmulcan2.x | β’ (π β π β π) |
lvecmulcan2.n | β’ (π β π β 0 ) |
Ref | Expression |
---|---|
lvecvscan2 | β’ (π β ((π΄ Β· π) = (π΅ Β· π) β π΄ = π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecmulcan2.n | . . . . 5 β’ (π β π β 0 ) | |
2 | 1 | neneqd 2946 | . . . 4 β’ (π β Β¬ π = 0 ) |
3 | biorf 936 | . . . . 5 β’ (Β¬ π = 0 β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β (π = 0 β¨ (π΄(-gβπΉ)π΅) = (0gβπΉ)))) | |
4 | orcom 869 | . . . . 5 β’ ((π = 0 β¨ (π΄(-gβπΉ)π΅) = (0gβπΉ)) β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β¨ π = 0 )) | |
5 | 3, 4 | bitrdi 287 | . . . 4 β’ (Β¬ π = 0 β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β¨ π = 0 ))) |
6 | 2, 5 | syl 17 | . . 3 β’ (π β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β¨ π = 0 ))) |
7 | lvecmulcan2.v | . . . 4 β’ π = (Baseβπ) | |
8 | lvecmulcan2.s | . . . 4 β’ Β· = ( Β·π βπ) | |
9 | lvecmulcan2.f | . . . 4 β’ πΉ = (Scalarβπ) | |
10 | lvecmulcan2.k | . . . 4 β’ πΎ = (BaseβπΉ) | |
11 | eqid 2733 | . . . 4 β’ (0gβπΉ) = (0gβπΉ) | |
12 | lvecmulcan2.o | . . . 4 β’ 0 = (0gβπ) | |
13 | lvecmulcan2.w | . . . 4 β’ (π β π β LVec) | |
14 | lveclmod 20717 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
15 | 13, 14 | syl 17 | . . . . . 6 β’ (π β π β LMod) |
16 | 9 | lmodfgrp 20480 | . . . . . 6 β’ (π β LMod β πΉ β Grp) |
17 | 15, 16 | syl 17 | . . . . 5 β’ (π β πΉ β Grp) |
18 | lvecmulcan2.a | . . . . 5 β’ (π β π΄ β πΎ) | |
19 | lvecmulcan2.b | . . . . 5 β’ (π β π΅ β πΎ) | |
20 | eqid 2733 | . . . . . 6 β’ (-gβπΉ) = (-gβπΉ) | |
21 | 10, 20 | grpsubcl 18903 | . . . . 5 β’ ((πΉ β Grp β§ π΄ β πΎ β§ π΅ β πΎ) β (π΄(-gβπΉ)π΅) β πΎ) |
22 | 17, 18, 19, 21 | syl3anc 1372 | . . . 4 β’ (π β (π΄(-gβπΉ)π΅) β πΎ) |
23 | lvecmulcan2.x | . . . 4 β’ (π β π β π) | |
24 | 7, 8, 9, 10, 11, 12, 13, 22, 23 | lvecvs0or 20721 | . . 3 β’ (π β (((π΄(-gβπΉ)π΅) Β· π) = 0 β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β¨ π = 0 ))) |
25 | eqid 2733 | . . . . 5 β’ (-gβπ) = (-gβπ) | |
26 | 7, 8, 9, 10, 25, 20, 15, 18, 19, 23 | lmodsubdir 20530 | . . . 4 β’ (π β ((π΄(-gβπΉ)π΅) Β· π) = ((π΄ Β· π)(-gβπ)(π΅ Β· π))) |
27 | 26 | eqeq1d 2735 | . . 3 β’ (π β (((π΄(-gβπΉ)π΅) Β· π) = 0 β ((π΄ Β· π)(-gβπ)(π΅ Β· π)) = 0 )) |
28 | 6, 24, 27 | 3bitr2rd 308 | . 2 β’ (π β (((π΄ Β· π)(-gβπ)(π΅ Β· π)) = 0 β (π΄(-gβπΉ)π΅) = (0gβπΉ))) |
29 | 7, 9, 8, 10 | lmodvscl 20489 | . . . 4 β’ ((π β LMod β§ π΄ β πΎ β§ π β π) β (π΄ Β· π) β π) |
30 | 15, 18, 23, 29 | syl3anc 1372 | . . 3 β’ (π β (π΄ Β· π) β π) |
31 | 7, 9, 8, 10 | lmodvscl 20489 | . . . 4 β’ ((π β LMod β§ π΅ β πΎ β§ π β π) β (π΅ Β· π) β π) |
32 | 15, 19, 23, 31 | syl3anc 1372 | . . 3 β’ (π β (π΅ Β· π) β π) |
33 | 7, 12, 25 | lmodsubeq0 20531 | . . 3 β’ ((π β LMod β§ (π΄ Β· π) β π β§ (π΅ Β· π) β π) β (((π΄ Β· π)(-gβπ)(π΅ Β· π)) = 0 β (π΄ Β· π) = (π΅ Β· π))) |
34 | 15, 30, 32, 33 | syl3anc 1372 | . 2 β’ (π β (((π΄ Β· π)(-gβπ)(π΅ Β· π)) = 0 β (π΄ Β· π) = (π΅ Β· π))) |
35 | 10, 11, 20 | grpsubeq0 18909 | . . 3 β’ ((πΉ β Grp β§ π΄ β πΎ β§ π΅ β πΎ) β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β π΄ = π΅)) |
36 | 17, 18, 19, 35 | syl3anc 1372 | . 2 β’ (π β ((π΄(-gβπΉ)π΅) = (0gβπΉ) β π΄ = π΅)) |
37 | 28, 34, 36 | 3bitr3d 309 | 1 β’ (π β ((π΄ Β· π) = (π΅ Β· π) β π΄ = π΅)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β¨ wo 846 = wceq 1542 β wcel 2107 β wne 2941 βcfv 6544 (class class class)co 7409 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 0gc0g 17385 Grpcgrp 18819 -gcsg 18821 LModclmod 20471 LVecclvec 20713 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mgp 19988 df-ur 20005 df-ring 20058 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-drng 20359 df-lmod 20473 df-lvec 20714 |
This theorem is referenced by: lspsneu 20736 lvecindp 20751 lvecindp2 20752 linds2eq 32497 lshpsmreu 37979 lshpkrlem5 37984 hgmapval1 40764 hgmapadd 40765 hgmapmul 40766 hgmaprnlem1N 40767 hgmap11 40773 |
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