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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 20354 | . . . 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 1372 | . . 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 2733 | . . . 4 β’ (1rβπΉ) = (1rβπΉ) | |
15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 20392 | . . 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 1372 | . 2 β’ (π β (π β (π΄ Β· π)) = (π + ((πβ(1rβπΉ)) Β· (π΄ Β· π)))) |
17 | 6 | lmodring 20344 | . . . . . . . 8 β’ (π β LMod β πΉ β Ring) |
18 | 1, 17 | syl 17 | . . . . . . 7 β’ (π β πΉ β Ring) |
19 | ringgrp 19974 | . . . . . . 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 19994 | . . . . . . 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 18803 | . . . . . 6 β’ ((πΉ β Grp β§ (1rβπΉ) β πΎ) β (πβ(1rβπΉ)) β πΎ) |
24 | 20, 22, 23 | syl2anc 585 | . . . . 5 β’ (π β (πβ(1rβπΉ)) β πΎ) |
25 | eqid 2733 | . . . . . 6 β’ (.rβπΉ) = (.rβπΉ) | |
26 | 5, 6, 7, 8, 25 | lmodvsass 20362 | . . . . 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 1373 | . . . 4 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π΄) Β· π) = ((πβ(1rβπΉ)) Β· (π΄ Β· π))) |
28 | 8, 25, 14, 13, 18, 3 | ringnegl 20023 | . . . . 5 β’ (π β ((πβ(1rβπΉ))(.rβπΉ)π΄) = (πβπ΄)) |
29 | 28 | oveq1d 7373 | . . . 4 β’ (π β (((πβ(1rβπΉ))(.rβπΉ)π΄) Β· π) = ((πβπ΄) Β· π)) |
30 | 27, 29 | eqtr3d 2775 | . . 3 β’ (π β ((πβ(1rβπΉ)) Β· (π΄ Β· π)) = ((πβπ΄) Β· π)) |
31 | 30 | oveq2d 7374 | . 2 β’ (π β (π + ((πβ(1rβπΉ)) Β· (π΄ Β· π))) = (π + ((πβπ΄) Β· π))) |
32 | 16, 31 | eqtrd 2773 | 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 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 .rcmulr 17139 Scalarcsca 17141 Β·π cvsca 17142 Grpcgrp 18753 invgcminusg 18754 -gcsg 18755 1rcur 19918 Ringcrg 19969 LModclmod 20336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 |
This theorem is referenced by: lspexch 20606 baerlem5alem1 40217 baerlem5blem1 40218 |
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