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Mirrors > Home > MPE Home > Th. List > lvecprop2d | Structured version Visualization version GIF version |
Description: If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 21054 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.) |
Ref | Expression |
---|---|
lvecprop2d.b1 | β’ (π β π΅ = (BaseβπΎ)) |
lvecprop2d.b2 | β’ (π β π΅ = (BaseβπΏ)) |
lvecprop2d.f | β’ πΉ = (ScalarβπΎ) |
lvecprop2d.g | β’ πΊ = (ScalarβπΏ) |
lvecprop2d.p1 | β’ (π β π = (BaseβπΉ)) |
lvecprop2d.p2 | β’ (π β π = (BaseβπΊ)) |
lvecprop2d.1 | β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) |
lvecprop2d.2 | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβπΉ)π¦) = (π₯(+gβπΊ)π¦)) |
lvecprop2d.3 | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(.rβπΉ)π¦) = (π₯(.rβπΊ)π¦)) |
lvecprop2d.4 | β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) |
Ref | Expression |
---|---|
lvecprop2d | β’ (π β (πΎ β LVec β πΏ β LVec)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecprop2d.b1 | . . . 4 β’ (π β π΅ = (BaseβπΎ)) | |
2 | lvecprop2d.b2 | . . . 4 β’ (π β π΅ = (BaseβπΏ)) | |
3 | lvecprop2d.f | . . . 4 β’ πΉ = (ScalarβπΎ) | |
4 | lvecprop2d.g | . . . 4 β’ πΊ = (ScalarβπΏ) | |
5 | lvecprop2d.p1 | . . . 4 β’ (π β π = (BaseβπΉ)) | |
6 | lvecprop2d.p2 | . . . 4 β’ (π β π = (BaseβπΊ)) | |
7 | lvecprop2d.1 | . . . 4 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) | |
8 | lvecprop2d.2 | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβπΉ)π¦) = (π₯(+gβπΊ)π¦)) | |
9 | lvecprop2d.3 | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(.rβπΉ)π¦) = (π₯(.rβπΊ)π¦)) | |
10 | lvecprop2d.4 | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | lmodprop2d 20806 | . . 3 β’ (π β (πΎ β LMod β πΏ β LMod)) |
12 | 5, 6, 8, 9 | drngpropd 20660 | . . 3 β’ (π β (πΉ β DivRing β πΊ β DivRing)) |
13 | 11, 12 | anbi12d 630 | . 2 β’ (π β ((πΎ β LMod β§ πΉ β DivRing) β (πΏ β LMod β§ πΊ β DivRing))) |
14 | 3 | islvec 20988 | . 2 β’ (πΎ β LVec β (πΎ β LMod β§ πΉ β DivRing)) |
15 | 4 | islvec 20988 | . 2 β’ (πΏ β LVec β (πΏ β LMod β§ πΊ β DivRing)) |
16 | 13, 14, 15 | 3bitr4g 313 | 1 β’ (π β (πΎ β LVec β πΏ β LVec)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7413 Basecbs 17174 +gcplusg 17227 .rcmulr 17228 Scalarcsca 17230 Β·π cvsca 17231 DivRingcdr 20623 LModclmod 20742 LVecclvec 20986 |
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-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18892 df-mgp 20074 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-drng 20625 df-lmod 20744 df-lvec 20987 |
This theorem is referenced by: hlhillvec 41480 |
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