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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia1dim2 | Structured version Visualization version GIF version |
Description: Two expressions for a 1-dimensional subspace of partial vector space A (when πΉ is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Ref | Expression |
---|---|
dia1dim2.h | β’ π» = (LHypβπΎ) |
dia1dim2.t | β’ π = ((LTrnβπΎ)βπ) |
dia1dim2.r | β’ π = ((trLβπΎ)βπ) |
dva1dim2.u | β’ π = ((DVecAβπΎ)βπ) |
dia1dim2.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
dva1dim2.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
dia1dim2 | β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΌβ(π βπΉ)) = (πβ{πΉ})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia1dim2.h | . . . . . . 7 β’ π» = (LHypβπΎ) | |
2 | eqid 2737 | . . . . . . 7 β’ ((TEndoβπΎ)βπ) = ((TEndoβπΎ)βπ) | |
3 | dva1dim2.u | . . . . . . 7 β’ π = ((DVecAβπΎ)βπ) | |
4 | eqid 2737 | . . . . . . 7 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2737 | . . . . . . 7 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
6 | 1, 2, 3, 4, 5 | dvabase 39473 | . . . . . 6 β’ ((πΎ β HL β§ π β π») β (Baseβ(Scalarβπ)) = ((TEndoβπΎ)βπ)) |
7 | 6 | adantr 482 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (Baseβ(Scalarβπ)) = ((TEndoβπΎ)βπ)) |
8 | 7 | rexeqdv 3315 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (βπ β (Baseβ(Scalarβπ))π = (π ( Β·π βπ)πΉ) β βπ β ((TEndoβπΎ)βπ)π = (π ( Β·π βπ)πΉ))) |
9 | dia1dim2.t | . . . . . . . 8 β’ π = ((LTrnβπΎ)βπ) | |
10 | eqid 2737 | . . . . . . . 8 β’ ( Β·π βπ) = ( Β·π βπ) | |
11 | 1, 9, 2, 3, 10 | dvavsca 39483 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ (π β ((TEndoβπΎ)βπ) β§ πΉ β π)) β (π ( Β·π βπ)πΉ) = (π βπΉ)) |
12 | 11 | anass1rs 654 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π) β§ π β ((TEndoβπΎ)βπ)) β (π ( Β·π βπ)πΉ) = (π βπΉ)) |
13 | 12 | eqeq2d 2748 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π) β§ π β ((TEndoβπΎ)βπ)) β (π = (π ( Β·π βπ)πΉ) β π = (π βπΉ))) |
14 | 13 | rexbidva 3174 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (βπ β ((TEndoβπΎ)βπ)π = (π ( Β·π βπ)πΉ) β βπ β ((TEndoβπΎ)βπ)π = (π βπΉ))) |
15 | 8, 14 | bitrd 279 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (βπ β (Baseβ(Scalarβπ))π = (π ( Β·π βπ)πΉ) β βπ β ((TEndoβπΎ)βπ)π = (π βπΉ))) |
16 | 15 | abbidv 2806 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β {π β£ βπ β (Baseβ(Scalarβπ))π = (π ( Β·π βπ)πΉ)} = {π β£ βπ β ((TEndoβπΎ)βπ)π = (π βπΉ)}) |
17 | 1, 3 | dvalvec 39492 | . . . . 5 β’ ((πΎ β HL β§ π β π») β π β LVec) |
18 | 17 | adantr 482 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β π β LVec) |
19 | lveclmod 20570 | . . . 4 β’ (π β LVec β π β LMod) | |
20 | 18, 19 | syl 17 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β π β LMod) |
21 | simpr 486 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΉ β π) | |
22 | eqid 2737 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
23 | 1, 9, 3, 22 | dvavbase 39479 | . . . . 5 β’ ((πΎ β HL β§ π β π») β (Baseβπ) = π) |
24 | 23 | adantr 482 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (Baseβπ) = π) |
25 | 21, 24 | eleqtrrd 2841 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΉ β (Baseβπ)) |
26 | dva1dim2.n | . . . 4 β’ π = (LSpanβπ) | |
27 | 4, 5, 22, 10, 26 | lspsn 20466 | . . 3 β’ ((π β LMod β§ πΉ β (Baseβπ)) β (πβ{πΉ}) = {π β£ βπ β (Baseβ(Scalarβπ))π = (π ( Β·π βπ)πΉ)}) |
28 | 20, 25, 27 | syl2anc 585 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πβ{πΉ}) = {π β£ βπ β (Baseβ(Scalarβπ))π = (π ( Β·π βπ)πΉ)}) |
29 | dia1dim2.r | . . 3 β’ π = ((trLβπΎ)βπ) | |
30 | dia1dim2.i | . . 3 β’ πΌ = ((DIsoAβπΎ)βπ) | |
31 | 1, 9, 29, 2, 30 | dia1dim 39527 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΌβ(π βπΉ)) = {π β£ βπ β ((TEndoβπΎ)βπ)π = (π βπΉ)}) |
32 | 16, 28, 31 | 3eqtr4rd 2788 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΌβ(π βπΉ)) = (πβ{πΉ})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {cab 2714 βwrex 3074 {csn 4587 βcfv 6497 (class class class)co 7358 Basecbs 17084 Scalarcsca 17137 Β·π cvsca 17138 LModclmod 20325 LSpanclspn 20435 LVecclvec 20566 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 trLctrl 38624 TEndoctendo 39218 DVecAcdveca 39468 DIsoAcdia 39494 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-riotaBAD 37418 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 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-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 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-tpos 8158 df-undef 8205 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-0g 17324 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-sbg 18754 df-cmn 19565 df-abl 19566 df-mgp 19898 df-ur 19915 df-ring 19967 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-drng 20188 df-lmod 20327 df-lss 20396 df-lsp 20436 df-lvec 20567 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-tgrp 39209 df-tendo 39221 df-edring 39223 df-dveca 39469 df-disoa 39495 |
This theorem is referenced by: dia1dimid 39529 dia2dimlem5 39534 dia2dimlem10 39539 |
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