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Mirrors > Home > MPE Home > Th. List > lmodvsneg | Structured version Visualization version GIF version |
Description: Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
lmodvsneg.b | β’ π΅ = (Baseβπ) |
lmodvsneg.f | β’ πΉ = (Scalarβπ) |
lmodvsneg.s | β’ Β· = ( Β·π βπ) |
lmodvsneg.n | β’ π = (invgβπ) |
lmodvsneg.k | β’ πΎ = (BaseβπΉ) |
lmodvsneg.m | β’ π = (invgβπΉ) |
lmodvsneg.w | β’ (π β π β LMod) |
lmodvsneg.x | β’ (π β π β π΅) |
lmodvsneg.r | β’ (π β π β πΎ) |
Ref | Expression |
---|---|
lmodvsneg | β’ (π β (πβ(π Β· π)) = ((πβπ ) Β· π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsneg.w | . . 3 β’ (π β π β LMod) | |
2 | lmodvsneg.f | . . . . . . 7 β’ πΉ = (Scalarβπ) | |
3 | 2 | lmodring 20710 | . . . . . 6 β’ (π β LMod β πΉ β Ring) |
4 | 1, 3 | syl 17 | . . . . 5 β’ (π β πΉ β Ring) |
5 | ringgrp 20139 | . . . . 5 β’ (πΉ β Ring β πΉ β Grp) | |
6 | 4, 5 | syl 17 | . . . 4 β’ (π β πΉ β Grp) |
7 | lmodvsneg.k | . . . . . 6 β’ πΎ = (BaseβπΉ) | |
8 | eqid 2724 | . . . . . 6 β’ (1rβπΉ) = (1rβπΉ) | |
9 | 7, 8 | ringidcl 20161 | . . . . 5 β’ (πΉ β Ring β (1rβπΉ) β πΎ) |
10 | 4, 9 | syl 17 | . . . 4 β’ (π β (1rβπΉ) β πΎ) |
11 | lmodvsneg.m | . . . . 5 β’ π = (invgβπΉ) | |
12 | 7, 11 | grpinvcl 18913 | . . . 4 β’ ((πΉ β Grp β§ (1rβπΉ) β πΎ) β (πβ(1rβπΉ)) β πΎ) |
13 | 6, 10, 12 | syl2anc 583 | . . 3 β’ (π β (πβ(1rβπΉ)) β πΎ) |
14 | lmodvsneg.r | . . 3 β’ (π β π β πΎ) | |
15 | lmodvsneg.x | . . 3 β’ (π β π β π΅) | |
16 | lmodvsneg.b | . . . 4 β’ π΅ = (Baseβπ) | |
17 | lmodvsneg.s | . . . 4 β’ Β· = ( Β·π βπ) | |
18 | eqid 2724 | . . . 4 β’ (.rβπΉ) = (.rβπΉ) | |
19 | 16, 2, 17, 7, 18 | lmodvsass 20729 | . . 3 β’ ((π β LMod β§ ((πβ(1rβπΉ)) β πΎ β§ π β πΎ β§ π β π΅)) β (((πβ(1rβπΉ))(.rβπΉ)π ) Β· π) = ((πβ(1rβπΉ)) Β· (π Β· π))) |
20 | 1, 13, 14, 15, 19 | syl13anc 1369 | . 2 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π ) Β· π) = ((πβ(1rβπΉ)) Β· (π Β· π))) |
21 | 7, 18, 8, 11, 4, 14 | ringnegl 20197 | . . 3 β’ (π β ((πβ(1rβπΉ))(.rβπΉ)π ) = (πβπ )) |
22 | 21 | oveq1d 7417 | . 2 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π ) Β· π) = ((πβπ ) Β· π)) |
23 | 16, 2, 17, 7 | lmodvscl 20720 | . . . 4 β’ ((π β LMod β§ π β πΎ β§ π β π΅) β (π Β· π) β π΅) |
24 | 1, 14, 15, 23 | syl3anc 1368 | . . 3 β’ (π β (π Β· π) β π΅) |
25 | lmodvsneg.n | . . . 4 β’ π = (invgβπ) | |
26 | 16, 25, 2, 17, 8, 11 | lmodvneg1 20747 | . . 3 β’ ((π β LMod β§ (π Β· π) β π΅) β ((πβ(1rβπΉ)) Β· (π Β· π)) = (πβ(π Β· π))) |
27 | 1, 24, 26 | syl2anc 583 | . 2 β’ (π β ((πβ(1rβπΉ)) Β· (π Β· π)) = (πβ(π Β· π))) |
28 | 20, 22, 27 | 3eqtr3rd 2773 | 1 β’ (π β (πβ(π Β· π)) = ((πβπ ) Β· π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 Basecbs 17149 .rcmulr 17203 Scalarcsca 17205 Β·π cvsca 17206 Grpcgrp 18859 invgcminusg 18860 1rcur 20082 Ringcrg 20134 LModclmod 20702 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-lmod 20704 |
This theorem is referenced by: lmodnegadd 20753 clmvsneg 24971 linds2eq 32993 baerlem5alem1 41083 lincext3 47386 lindslinindimp2lem4 47391 lincresunit3 47411 |
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