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Mirrors > Home > MPE Home > Th. List > lvecvscan2 | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan2 30057 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 2945 | . . . 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 20582 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
15 | 13, 14 | syl 17 | . . . . . 6 β’ (π β π β LMod) |
16 | 9 | lmodfgrp 20345 | . . . . . 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 18832 | . . . . 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 20585 | . . 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 20395 | . . . 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 20354 | . . . 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 20354 | . . . 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 20396 | . . 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 18838 | . . 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 2940 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 Grpcgrp 18753 -gcsg 18755 LModclmod 20336 LVecclvec 20578 |
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-1st 7922 df-2nd 7923 df-tpos 8158 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-3 12222 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-drng 20199 df-lmod 20338 df-lvec 20579 |
This theorem is referenced by: lspsneu 20600 lvecindp 20615 lvecindp2 20616 linds2eq 32216 lshpsmreu 37617 lshpkrlem5 37622 hgmapval1 40402 hgmapadd 40403 hgmapmul 40404 hgmaprnlem1N 40405 hgmap11 40411 |
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