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Mirrors > Home > MPE Home > Th. List > lsppr | Structured version Visualization version GIF version |
Description: Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.) |
Ref | Expression |
---|---|
lsppr.v | β’ π = (Baseβπ) |
lsppr.a | β’ + = (+gβπ) |
lsppr.f | β’ πΉ = (Scalarβπ) |
lsppr.k | β’ πΎ = (BaseβπΉ) |
lsppr.t | β’ Β· = ( Β·π βπ) |
lsppr.n | β’ π = (LSpanβπ) |
lsppr.w | β’ (π β π β LMod) |
lsppr.x | β’ (π β π β π) |
lsppr.y | β’ (π β π β π) |
Ref | Expression |
---|---|
lsppr | β’ (π β (πβ{π, π}) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4627 | . . 3 β’ {π, π} = ({π} βͺ {π}) | |
2 | 1 | fveq2i 6894 | . 2 β’ (πβ{π, π}) = (πβ({π} βͺ {π})) |
3 | lsppr.w | . . . 4 β’ (π β π β LMod) | |
4 | lsppr.x | . . . . 5 β’ (π β π β π) | |
5 | 4 | snssd 4808 | . . . 4 β’ (π β {π} β π) |
6 | lsppr.y | . . . . 5 β’ (π β π β π) | |
7 | 6 | snssd 4808 | . . . 4 β’ (π β {π} β π) |
8 | lsppr.v | . . . . 5 β’ π = (Baseβπ) | |
9 | lsppr.n | . . . . 5 β’ π = (LSpanβπ) | |
10 | 8, 9 | lspun 20864 | . . . 4 β’ ((π β LMod β§ {π} β π β§ {π} β π) β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
11 | 3, 5, 7, 10 | syl3anc 1369 | . . 3 β’ (π β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
12 | eqid 2728 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
13 | 8, 12, 9 | lspsncl 20854 | . . . . 5 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
14 | 3, 4, 13 | syl2anc 583 | . . . 4 β’ (π β (πβ{π}) β (LSubSpβπ)) |
15 | 8, 12, 9 | lspsncl 20854 | . . . . 5 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
16 | 3, 6, 15 | syl2anc 583 | . . . 4 β’ (π β (πβ{π}) β (LSubSpβπ)) |
17 | eqid 2728 | . . . . 5 β’ (LSSumβπ) = (LSSumβπ) | |
18 | 12, 9, 17 | lsmsp 20964 | . . . 4 β’ ((π β LMod β§ (πβ{π}) β (LSubSpβπ) β§ (πβ{π}) β (LSubSpβπ)) β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
19 | 3, 14, 16, 18 | syl3anc 1369 | . . 3 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
20 | lsppr.a | . . . . 5 β’ + = (+gβπ) | |
21 | lsppr.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
22 | lsppr.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
23 | lsppr.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
24 | 8, 20, 21, 22, 23, 17, 9, 3, 4, 6 | lsmspsn 20962 | . . . 4 β’ (π β (π£ β ((πβ{π})(LSSumβπ)(πβ{π})) β βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π)))) |
25 | 24 | eqabdv 2863 | . . 3 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
26 | 11, 19, 25 | 3eqtr2d 2774 | . 2 β’ (π β (πβ({π} βͺ {π})) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
27 | 2, 26 | eqtrid 2780 | 1 β’ (π β (πβ{π, π}) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 {cab 2705 βwrex 3066 βͺ cun 3943 β wss 3945 {csn 4624 {cpr 4626 βcfv 6542 (class class class)co 7414 Basecbs 17173 +gcplusg 17226 Scalarcsca 17229 Β·π cvsca 17230 LSSumclsm 19582 LModclmod 20736 LSubSpclss 20808 LSpanclspn 20848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 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 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-ur 20115 df-ring 20168 df-lmod 20738 df-lss 20809 df-lsp 20849 |
This theorem is referenced by: lspprel 20972 |
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