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Mirrors > Home > MPE Home > Th. List > lvecvscan | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan 30834 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 2935 | . . . 4 β’ (π΄ β 0 β Β¬ π΄ = 0 ) | |
3 | biorf 933 | . . . 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 20954 | . . . 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 2726 | . . . 4 β’ (0gβπ) = (0gβπ) | |
13 | eqid 2726 | . . . 4 β’ (-gβπ) = (-gβπ) | |
14 | 11, 12, 13 | lmodsubeq0 20767 | . . 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 20765 | . . . 4 β’ (π β (π΄ Β· (π(-gβπ)π)) = ((π΄ Β· π)(-gβπ)(π΄ Β· π))) |
21 | 20 | eqeq1d 2728 | . . 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 20753 | . . . . 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 20959 | . . 3 β’ (π β ((π΄ Β· (π(-gβπ)π)) = (0gβπ) β (π΄ = 0 β¨ (π(-gβπ)π) = (0gβπ)))) |
26 | 11, 17, 16, 18 | lmodvscl 20724 | . . . . 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 20724 | . . . . 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 20767 | . . . 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 309 | . 2 β’ (π β ((π΄ = 0 β¨ (π(-gβπ)π) = (0gβπ)) β (π΄ Β· π) = (π΄ Β· π))) |
33 | 5, 15, 32 | 3bitr3rd 310 | 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 844 = wceq 1533 β wcel 2098 β wne 2934 βcfv 6537 (class class class)co 7405 Basecbs 17153 Scalarcsca 17209 Β·π cvsca 17210 0gc0g 17394 -gcsg 18865 LModclmod 20706 LVecclvec 20950 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-drng 20589 df-lmod 20708 df-lvec 20951 |
This theorem is referenced by: (None) |
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