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Mirrors > Home > MPE Home > Th. List > lvecvscan | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan 30926 analog.) (Contributed by NM, 2-Jul-2014.) |
Ref | Expression |
---|---|
lvecmulcan.v | β’ π = (Baseβπ) |
lvecmulcan.s | β’ Β· = ( Β·π βπ) |
lvecmulcan.f | β’ πΉ = (Scalarβπ) |
lvecmulcan.k | β’ πΎ = (BaseβπΉ) |
lvecmulcan.o | β’ 0 = (0gβπΉ) |
lvecmulcan.w | β’ (π β π β LVec) |
lvecmulcan.a | β’ (π β π΄ β πΎ) |
lvecmulcan.x | β’ (π β π β π) |
lvecmulcan.y | β’ (π β π β π) |
lvecmulcan.n | β’ (π β π΄ β 0 ) |
Ref | Expression |
---|---|
lvecvscan | β’ (π β ((π΄ Β· π) = (π΄ Β· π) β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecmulcan.n | . . 3 β’ (π β π΄ β 0 ) | |
2 | df-ne 2931 | . . . 4 β’ (π΄ β 0 β Β¬ π΄ = 0 ) | |
3 | biorf 934 | . . . 4 β’ (Β¬ π΄ = 0 β ((π(-gβπ)π) = (0gβπ) β (π΄ = 0 β¨ (π(-gβπ)π) = (0gβπ)))) | |
4 | 2, 3 | sylbi 216 | . . 3 β’ (π΄ β 0 β ((π(-gβπ)π) = (0gβπ) β (π΄ = 0 β¨ (π(-gβπ)π) = (0gβπ)))) |
5 | 1, 4 | syl 17 | . 2 β’ (π β ((π(-gβπ)π) = (0gβπ) β (π΄ = 0 β¨ (π(-gβπ)π) = (0gβπ)))) |
6 | lvecmulcan.w | . . . 4 β’ (π β π β LVec) | |
7 | lveclmod 20995 | . . . 4 β’ (π β LVec β π β LMod) | |
8 | 6, 7 | syl 17 | . . 3 β’ (π β π β LMod) |
9 | lvecmulcan.x | . . 3 β’ (π β π β π) | |
10 | lvecmulcan.y | . . 3 β’ (π β π β π) | |
11 | lvecmulcan.v | . . . 4 β’ π = (Baseβπ) | |
12 | eqid 2725 | . . . 4 β’ (0gβπ) = (0gβπ) | |
13 | eqid 2725 | . . . 4 β’ (-gβπ) = (-gβπ) | |
14 | 11, 12, 13 | lmodsubeq0 20808 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β ((π(-gβπ)π) = (0gβπ) β π = π)) |
15 | 8, 9, 10, 14 | syl3anc 1368 | . 2 β’ (π β ((π(-gβπ)π) = (0gβπ) β π = π)) |
16 | lvecmulcan.s | . . . . 5 β’ Β· = ( Β·π βπ) | |
17 | lvecmulcan.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
18 | lvecmulcan.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
19 | lvecmulcan.a | . . . . 5 β’ (π β π΄ β πΎ) | |
20 | 11, 16, 17, 18, 13, 8, 19, 9, 10 | lmodsubdi 20806 | . . . 4 β’ (π β (π΄ Β· (π(-gβπ)π)) = ((π΄ Β· π)(-gβπ)(π΄ Β· π))) |
21 | 20 | eqeq1d 2727 | . . 3 β’ (π β ((π΄ Β· (π(-gβπ)π)) = (0gβπ) β ((π΄ Β· π)(-gβπ)(π΄ Β· π)) = (0gβπ))) |
22 | lvecmulcan.o | . . . 4 β’ 0 = (0gβπΉ) | |
23 | 11, 13 | lmodvsubcl 20794 | . . . . 5 β’ ((π β LMod β§ π β π β§ π β π) β (π(-gβπ)π) β π) |
24 | 8, 9, 10, 23 | syl3anc 1368 | . . . 4 β’ (π β (π(-gβπ)π) β π) |
25 | 11, 16, 17, 18, 22, 12, 6, 19, 24 | lvecvs0or 21000 | . . 3 β’ (π β ((π΄ Β· (π(-gβπ)π)) = (0gβπ) β (π΄ = 0 β¨ (π(-gβπ)π) = (0gβπ)))) |
26 | 11, 17, 16, 18 | lmodvscl 20765 | . . . . 5 β’ ((π β LMod β§ π΄ β πΎ β§ π β π) β (π΄ Β· π) β π) |
27 | 8, 19, 9, 26 | syl3anc 1368 | . . . 4 β’ (π β (π΄ Β· π) β π) |
28 | 11, 17, 16, 18 | lmodvscl 20765 | . . . . 5 β’ ((π β LMod β§ π΄ β πΎ β§ π β π) β (π΄ Β· π) β π) |
29 | 8, 19, 10, 28 | syl3anc 1368 | . . . 4 β’ (π β (π΄ Β· π) β π) |
30 | 11, 12, 13 | lmodsubeq0 20808 | . . . 4 β’ ((π β LMod β§ (π΄ Β· π) β π β§ (π΄ Β· π) β π) β (((π΄ Β· π)(-gβπ)(π΄ Β· π)) = (0gβπ) β (π΄ Β· π) = (π΄ Β· π))) |
31 | 8, 27, 29, 30 | syl3anc 1368 | . . 3 β’ (π β (((π΄ Β· π)(-gβπ)(π΄ Β· π)) = (0gβπ) β (π΄ Β· π) = (π΄ Β· π))) |
32 | 21, 25, 31 | 3bitr3d 308 | . 2 β’ (π β ((π΄ = 0 β¨ (π(-gβπ)π) = (0gβπ)) β (π΄ Β· π) = (π΄ Β· π))) |
33 | 5, 15, 32 | 3bitr3rd 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 845 = wceq 1533 β wcel 2098 β wne 2930 βcfv 6543 (class class class)co 7416 Basecbs 17179 Scalarcsca 17235 Β·π cvsca 17236 0gc0g 17420 -gcsg 18896 LModclmod 20747 LVecclvec 20991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-drng 20630 df-lmod 20749 df-lvec 20992 |
This theorem is referenced by: (None) |
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