![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > lmodsubvs | GIF version |
Description: Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.) |
Ref | Expression |
---|---|
lmodsubvs.v | β’ π = (Baseβπ) |
lmodsubvs.p | β’ + = (+gβπ) |
lmodsubvs.m | β’ β = (-gβπ) |
lmodsubvs.t | β’ Β· = ( Β·π βπ) |
lmodsubvs.f | β’ πΉ = (Scalarβπ) |
lmodsubvs.k | β’ πΎ = (BaseβπΉ) |
lmodsubvs.n | β’ π = (invgβπΉ) |
lmodsubvs.w | β’ (π β π β LMod) |
lmodsubvs.a | β’ (π β π΄ β πΎ) |
lmodsubvs.x | β’ (π β π β π) |
lmodsubvs.y | β’ (π β π β π) |
Ref | Expression |
---|---|
lmodsubvs | β’ (π β (π β (π΄ Β· π)) = (π + ((πβπ΄) Β· π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodsubvs.w | . . 3 β’ (π β π β LMod) | |
2 | lmodsubvs.x | . . 3 β’ (π β π β π) | |
3 | lmodsubvs.a | . . . 4 β’ (π β π΄ β πΎ) | |
4 | lmodsubvs.y | . . . 4 β’ (π β π β π) | |
5 | lmodsubvs.v | . . . . 5 β’ π = (Baseβπ) | |
6 | lmodsubvs.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
7 | lmodsubvs.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
8 | lmodsubvs.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
9 | 5, 6, 7, 8 | lmodvscl 13588 | . . . 4 β’ ((π β LMod β§ π΄ β πΎ β§ π β π) β (π΄ Β· π) β π) |
10 | 1, 3, 4, 9 | syl3anc 1249 | . . 3 β’ (π β (π΄ Β· π) β π) |
11 | lmodsubvs.p | . . . 4 β’ + = (+gβπ) | |
12 | lmodsubvs.m | . . . 4 β’ β = (-gβπ) | |
13 | lmodsubvs.n | . . . 4 β’ π = (invgβπΉ) | |
14 | eqid 2189 | . . . 4 β’ (1rβπΉ) = (1rβπΉ) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 13625 | . . 3 β’ ((π β LMod β§ π β π β§ (π΄ Β· π) β π) β (π β (π΄ Β· π)) = (π + ((πβ(1rβπΉ)) Β· (π΄ Β· π)))) |
16 | 1, 2, 10, 15 | syl3anc 1249 | . 2 β’ (π β (π β (π΄ Β· π)) = (π + ((πβ(1rβπΉ)) Β· (π΄ Β· π)))) |
17 | 6 | lmodring 13578 | . . . . . . . 8 β’ (π β LMod β πΉ β Ring) |
18 | 1, 17 | syl 14 | . . . . . . 7 β’ (π β πΉ β Ring) |
19 | ringgrp 13322 | . . . . . . 7 β’ (πΉ β Ring β πΉ β Grp) | |
20 | 18, 19 | syl 14 | . . . . . 6 β’ (π β πΉ β Grp) |
21 | 8, 14 | ringidcl 13341 | . . . . . . 7 β’ (πΉ β Ring β (1rβπΉ) β πΎ) |
22 | 18, 21 | syl 14 | . . . . . 6 β’ (π β (1rβπΉ) β πΎ) |
23 | 8, 13 | grpinvcl 12964 | . . . . . 6 β’ ((πΉ β Grp β§ (1rβπΉ) β πΎ) β (πβ(1rβπΉ)) β πΎ) |
24 | 20, 22, 23 | syl2anc 411 | . . . . 5 β’ (π β (πβ(1rβπΉ)) β πΎ) |
25 | eqid 2189 | . . . . . 6 β’ (.rβπΉ) = (.rβπΉ) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 13596 | . . . . 5 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))) |
27 | 1, 24, 3, 4, 26 | syl13anc 1251 | . . . 4 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π΄) Β· π) = ((πβ(1rβπΉ)) Β· (π΄ Β· π))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 13370 | . . . . 5 β’ (π β ((πβ(1rβπΉ))(.rβπΉ)π΄) = (πβπ΄)) |
29 | 28 | oveq1d 5906 | . . . 4 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π΄) Β· π) = ((πβπ΄) Β· π)) |
30 | 27, 29 | eqtr3d 2224 | . . 3 β’ (π β ((πβ(1rβπΉ)) Β· (π΄ Β· π)) = ((πβπ΄) Β· π)) |
31 | 30 | oveq2d 5907 | . 2 β’ (π β (π + ((πβ(1rβπΉ)) Β· (π΄ Β· π))) = (π + ((πβπ΄) Β· π))) |
32 | 16, 31 | eqtrd 2222 | 1 β’ (π β (π β (π΄ Β· π)) = (π + ((πβπ΄) Β· π))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1364 β wcel 2160 βcfv 5231 (class class class)co 5891 Basecbs 12486 +gcplusg 12561 .rcmulr 12562 Scalarcsca 12564 Β·π cvsca 12565 Grpcgrp 12917 invgcminusg 12918 -gcsg 12919 1rcur 13280 Ringcrg 13317 LModclmod 13570 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-pre-ltirr 7942 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8013 df-mnf 8014 df-ltxr 8016 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-5 9000 df-6 9001 df-ndx 12489 df-slot 12490 df-base 12492 df-sets 12493 df-plusg 12574 df-mulr 12575 df-sca 12577 df-vsca 12578 df-0g 12735 df-mgm 12804 df-sgrp 12837 df-mnd 12850 df-grp 12920 df-minusg 12921 df-sbg 12922 df-mgp 13242 df-ur 13281 df-ring 13319 df-lmod 13572 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |