![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sralvec | Structured version Visualization version GIF version |
Description: Given a sub division ring πΉ of a division ring πΈ, πΈ may be considered as a vector space over πΉ, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.) |
Ref | Expression |
---|---|
sralvec.a | β’ π΄ = ((subringAlg βπΈ)βπ) |
sralvec.f | β’ πΉ = (πΈ βΎs π) |
Ref | Expression |
---|---|
sralvec | β’ ((πΈ β DivRing β§ πΉ β DivRing β§ π β (SubRingβπΈ)) β π΄ β LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sralvec.a | . . 3 β’ π΄ = ((subringAlg βπΈ)βπ) | |
2 | eqid 2733 | . . . . 5 β’ ((subringAlg βπΈ)βπ) = ((subringAlg βπΈ)βπ) | |
3 | 2 | sralmod 20701 | . . . 4 β’ (π β (SubRingβπΈ) β ((subringAlg βπΈ)βπ) β LMod) |
4 | 3 | 3ad2ant3 1136 | . . 3 β’ ((πΈ β DivRing β§ πΉ β DivRing β§ π β (SubRingβπΈ)) β ((subringAlg βπΈ)βπ) β LMod) |
5 | 1, 4 | eqeltrid 2838 | . 2 β’ ((πΈ β DivRing β§ πΉ β DivRing β§ π β (SubRingβπΈ)) β π΄ β LMod) |
6 | sralvec.f | . . . . 5 β’ πΉ = (πΈ βΎs π) | |
7 | 1 | a1i 11 | . . . . . 6 β’ (π β (SubRingβπΈ) β π΄ = ((subringAlg βπΈ)βπ)) |
8 | eqid 2733 | . . . . . . 7 β’ (BaseβπΈ) = (BaseβπΈ) | |
9 | 8 | subrgss 20265 | . . . . . 6 β’ (π β (SubRingβπΈ) β π β (BaseβπΈ)) |
10 | 7, 9 | srasca 20691 | . . . . 5 β’ (π β (SubRingβπΈ) β (πΈ βΎs π) = (Scalarβπ΄)) |
11 | 6, 10 | eqtrid 2785 | . . . 4 β’ (π β (SubRingβπΈ) β πΉ = (Scalarβπ΄)) |
12 | 11 | 3ad2ant3 1136 | . . 3 β’ ((πΈ β DivRing β§ πΉ β DivRing β§ π β (SubRingβπΈ)) β πΉ = (Scalarβπ΄)) |
13 | simp2 1138 | . . 3 β’ ((πΈ β DivRing β§ πΉ β DivRing β§ π β (SubRingβπΈ)) β πΉ β DivRing) | |
14 | 12, 13 | eqeltrrd 2835 | . 2 β’ ((πΈ β DivRing β§ πΉ β DivRing β§ π β (SubRingβπΈ)) β (Scalarβπ΄) β DivRing) |
15 | eqid 2733 | . . 3 β’ (Scalarβπ΄) = (Scalarβπ΄) | |
16 | 15 | islvec 20609 | . 2 β’ (π΄ β LVec β (π΄ β LMod β§ (Scalarβπ΄) β DivRing)) |
17 | 5, 14, 16 | sylanbrc 584 | 1 β’ ((πΈ β DivRing β§ πΉ β DivRing β§ π β (SubRingβπΈ)) β π΄ β LVec) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 Basecbs 17091 βΎs cress 17120 Scalarcsca 17144 DivRingcdr 20219 SubRingcsubrg 20260 LModclmod 20365 LVecclvec 20607 subringAlg csra 20674 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-subg 18933 df-mgp 19905 df-ur 19922 df-ring 19974 df-subrg 20262 df-lmod 20367 df-lvec 20608 df-sra 20678 |
This theorem is referenced by: srafldlvec 32352 drgextgsum 32358 rgmoddim 32369 fedgmullem1 32388 fedgmullem2 32389 fedgmul 32390 fldextsralvec 32408 extdgcl 32409 extdggt0 32410 |
Copyright terms: Public domain | W3C validator |