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Mirrors > Home > ILE Home > Th. List > lmodvsneg | 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 13448 | . . . . . 6 β’ (π β LMod β πΉ β Ring) |
4 | 1, 3 | syl 14 | . . . . 5 β’ (π β πΉ β Ring) |
5 | ringgrp 13238 | . . . . 5 β’ (πΉ β Ring β πΉ β Grp) | |
6 | 4, 5 | syl 14 | . . . 4 β’ (π β πΉ β Grp) |
7 | lmodvsneg.k | . . . . . 6 β’ πΎ = (BaseβπΉ) | |
8 | eqid 2187 | . . . . . 6 β’ (1rβπΉ) = (1rβπΉ) | |
9 | 7, 8 | ringidcl 13257 | . . . . 5 β’ (πΉ β Ring β (1rβπΉ) β πΎ) |
10 | 4, 9 | syl 14 | . . . 4 β’ (π β (1rβπΉ) β πΎ) |
11 | lmodvsneg.m | . . . . 5 β’ π = (invgβπΉ) | |
12 | 7, 11 | grpinvcl 12942 | . . . 4 β’ ((πΉ β Grp β§ (1rβπΉ) β πΎ) β (πβ(1rβπΉ)) β πΎ) |
13 | 6, 10, 12 | syl2anc 411 | . . 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 2187 | . . . 4 β’ (.rβπΉ) = (.rβπΉ) | |
19 | 16, 2, 17, 7, 18 | lmodvsass 13466 | . . 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 1250 | . 2 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π ) Β· π) = ((πβ(1rβπΉ)) Β· (π Β· π))) |
21 | 7, 18, 8, 11, 4, 14 | ringnegl 13286 | . . 3 β’ (π β ((πβ(1rβπΉ))(.rβπΉ)π ) = (πβπ )) |
22 | 21 | oveq1d 5903 | . 2 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π ) Β· π) = ((πβπ ) Β· π)) |
23 | 16, 2, 17, 7 | lmodvscl 13458 | . . . 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 1248 | . . 3 β’ (π β (π Β· π) β π΅) |
25 | lmodvsneg.n | . . . 4 β’ π = (invgβπ) | |
26 | 16, 25, 2, 17, 8, 11 | lmodvneg1 13483 | . . 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 411 | . 2 β’ (π β ((πβ(1rβπΉ)) Β· (π Β· π)) = (πβ(π Β· π))) |
28 | 20, 22, 27 | 3eqtr3rd 2229 | 1 β’ (π β (πβ(π Β· π)) = ((πβπ ) Β· π)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1363 β wcel 2158 βcfv 5228 (class class class)co 5888 Basecbs 12475 .rcmulr 12551 Scalarcsca 12553 Β·π cvsca 12554 Grpcgrp 12896 invgcminusg 12897 1rcur 13196 Ringcrg 13233 LModclmod 13440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltirr 7936 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-plusg 12563 df-mulr 12564 df-sca 12566 df-vsca 12567 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12837 df-grp 12899 df-minusg 12900 df-mgp 13163 df-ur 13197 df-ring 13235 df-lmod 13442 |
This theorem is referenced by: lmodnegadd 13489 |
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