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Mirrors > Home > MPE Home > Th. List > lmodsubvs | Structured version Visualization version 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 20765 | . . . 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 1368 | . . 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 2725 | . . . 4 β’ (1rβπΉ) = (1rβπΉ) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 20804 | . . 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 1368 | . 2 β’ (π β (π β (π΄ Β· π)) = (π + ((πβ(1rβπΉ)) Β· (π΄ Β· π)))) |
17 | 6 | lmodring 20755 | . . . . . . . 8 β’ (π β LMod β πΉ β Ring) |
18 | 1, 17 | syl 17 | . . . . . . 7 β’ (π β πΉ β Ring) |
19 | ringgrp 20182 | . . . . . . 7 β’ (πΉ β Ring β πΉ β Grp) | |
20 | 18, 19 | syl 17 | . . . . . 6 β’ (π β πΉ β Grp) |
21 | 8, 14 | ringidcl 20206 | . . . . . . 7 β’ (πΉ β Ring β (1rβπΉ) β πΎ) |
22 | 18, 21 | syl 17 | . . . . . 6 β’ (π β (1rβπΉ) β πΎ) |
23 | 8, 13 | grpinvcl 18948 | . . . . . 6 β’ ((πΉ β Grp β§ (1rβπΉ) β πΎ) β (πβ(1rβπΉ)) β πΎ) |
24 | 20, 22, 23 | syl2anc 582 | . . . . 5 β’ (π β (πβ(1rβπΉ)) β πΎ) |
25 | eqid 2725 | . . . . . 6 β’ (.rβπΉ) = (.rβπΉ) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 20774 | . . . . 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 1369 | . . . 4 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π΄) Β· π) = ((πβ(1rβπΉ)) Β· (π΄ Β· π))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 20242 | . . . . 5 β’ (π β ((πβ(1rβπΉ))(.rβπΉ)π΄) = (πβπ΄)) |
29 | 28 | oveq1d 7431 | . . . 4 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π΄) Β· π) = ((πβπ΄) Β· π)) |
30 | 27, 29 | eqtr3d 2767 | . . 3 β’ (π β ((πβ(1rβπΉ)) Β· (π΄ Β· π)) = ((πβπ΄) Β· π)) |
31 | 30 | oveq2d 7432 | . 2 β’ (π β (π + ((πβ(1rβπΉ)) Β· (π΄ Β· π))) = (π + ((πβπ΄) Β· π))) |
32 | 16, 31 | eqtrd 2765 | 1 β’ (π β (π β (π΄ Β· π)) = (π + ((πβπ΄) Β· π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 .rcmulr 17233 Scalarcsca 17235 Β·π cvsca 17236 Grpcgrp 18894 invgcminusg 18895 -gcsg 18896 1rcur 20125 Ringcrg 20177 LModclmod 20747 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-lmod 20749 |
This theorem is referenced by: lspexch 21021 baerlem5alem1 41237 baerlem5blem1 41238 |
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