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Mirrors > Home > MPE Home > Th. List > clmvs2 | Structured version Visualization version GIF version |
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.) |
Ref | Expression |
---|---|
clmvs1.v | β’ π = (Baseβπ) |
clmvs1.s | β’ Β· = ( Β·π βπ) |
clmvs2.a | β’ + = (+gβπ) |
Ref | Expression |
---|---|
clmvs2 | β’ ((π β βMod β§ π΄ β π) β (π΄ + π΄) = (2 Β· π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 12082 | . . . 4 β’ 2 = (1 + 1) | |
2 | 1 | oveq1i 7317 | . . 3 β’ (2 Β· π΄) = ((1 + 1) Β· π΄) |
3 | 2 | a1i 11 | . 2 β’ ((π β βMod β§ π΄ β π) β (2 Β· π΄) = ((1 + 1) Β· π΄)) |
4 | simpl 484 | . . 3 β’ ((π β βMod β§ π΄ β π) β π β βMod) | |
5 | eqid 2736 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
6 | 5 | clm1 24281 | . . . . 5 β’ (π β βMod β 1 = (1rβ(Scalarβπ))) |
7 | clmlmod 24275 | . . . . . 6 β’ (π β βMod β π β LMod) | |
8 | eqid 2736 | . . . . . . 7 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
9 | eqid 2736 | . . . . . . 7 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
10 | 5, 8, 9 | lmod1cl 20195 | . . . . . 6 β’ (π β LMod β (1rβ(Scalarβπ)) β (Baseβ(Scalarβπ))) |
11 | 7, 10 | syl 17 | . . . . 5 β’ (π β βMod β (1rβ(Scalarβπ)) β (Baseβ(Scalarβπ))) |
12 | 6, 11 | eqeltrd 2837 | . . . 4 β’ (π β βMod β 1 β (Baseβ(Scalarβπ))) |
13 | 12 | adantr 482 | . . 3 β’ ((π β βMod β§ π΄ β π) β 1 β (Baseβ(Scalarβπ))) |
14 | simpr 486 | . . 3 β’ ((π β βMod β§ π΄ β π) β π΄ β π) | |
15 | clmvs1.v | . . . 4 β’ π = (Baseβπ) | |
16 | clmvs1.s | . . . 4 β’ Β· = ( Β·π βπ) | |
17 | clmvs2.a | . . . 4 β’ + = (+gβπ) | |
18 | 15, 5, 16, 8, 17 | clmvsdir 24299 | . . 3 β’ ((π β βMod β§ (1 β (Baseβ(Scalarβπ)) β§ 1 β (Baseβ(Scalarβπ)) β§ π΄ β π)) β ((1 + 1) Β· π΄) = ((1 Β· π΄) + (1 Β· π΄))) |
19 | 4, 13, 13, 14, 18 | syl13anc 1372 | . 2 β’ ((π β βMod β§ π΄ β π) β ((1 + 1) Β· π΄) = ((1 Β· π΄) + (1 Β· π΄))) |
20 | 15, 16 | clmvs1 24301 | . . 3 β’ ((π β βMod β§ π΄ β π) β (1 Β· π΄) = π΄) |
21 | 20, 20 | oveq12d 7325 | . 2 β’ ((π β βMod β§ π΄ β π) β ((1 Β· π΄) + (1 Β· π΄)) = (π΄ + π΄)) |
22 | 3, 19, 21 | 3eqtrrd 2781 | 1 β’ ((π β βMod β§ π΄ β π) β (π΄ + π΄) = (2 Β· π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 βcfv 6458 (class class class)co 7307 1c1 10918 + caddc 10920 2c2 12074 Basecbs 16957 +gcplusg 17007 Scalarcsca 17010 Β·π cvsca 17011 1rcur 19782 LModclmod 20168 βModcclm 24270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-addf 10996 ax-mulf 10997 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-fz 13286 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-0g 17197 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-grp 18625 df-subg 18797 df-cmn 19433 df-mgp 19766 df-ur 19783 df-ring 19830 df-cring 19831 df-subrg 20067 df-lmod 20170 df-cnfld 20643 df-clm 24271 |
This theorem is referenced by: (None) |
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