Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dihprrn | Structured version Visualization version GIF version | ||
| Description: The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.) |
| Ref | Expression |
|---|---|
| dihprrn.h | β’ π» = (LHypβπΎ) |
| dihprrn.u | β’ π = ((DVecHβπΎ)βπ) |
| dihprrn.v | β’ π = (Baseβπ) |
| dihprrn.n | β’ π = (LSpanβπ) |
| dihprrn.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
| dihprrn.k | β’ (π β (πΎ β HL β§ π β π»)) |
| dihprrn.x | β’ (π β π β π) |
| dihprrn.y | β’ (π β π β π) |
| Ref | Expression |
|---|---|
| dihprrn | β’ (π β (πβ{π, π}) β ran πΌ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prcom 4731 | . . . . . 6 β’ {π, π} = {π, π} | |
| 2 | preq2 4733 | . . . . . 6 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) | |
| 3 | 1, 2 | eqtrid 2778 | . . . . 5 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) |
| 4 | 3 | fveq2d 6888 | . . . 4 β’ (π = (0gβπ) β (πβ{π, π}) = (πβ{π, (0gβπ)})) |
| 5 | dihprrn.v | . . . . 5 β’ π = (Baseβπ) | |
| 6 | eqid 2726 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
| 7 | dihprrn.n | . . . . 5 β’ π = (LSpanβπ) | |
| 8 | dihprrn.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
| 9 | dihprrn.u | . . . . . 6 β’ π = ((DVecHβπΎ)βπ) | |
| 10 | dihprrn.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
| 11 | 8, 9, 10 | dvhlmod 40493 | . . . . 5 β’ (π β π β LMod) |
| 12 | dihprrn.y | . . . . 5 β’ (π β π β π) | |
| 13 | 5, 6, 7, 11, 12 | lsppr0 20937 | . . . 4 β’ (π β (πβ{π, (0gβπ)}) = (πβ{π})) |
| 14 | 4, 13 | sylan9eqr 2788 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) = (πβ{π})) |
| 15 | dihprrn.i | . . . . . 6 β’ πΌ = ((DIsoHβπΎ)βπ) | |
| 16 | 8, 9, 5, 7, 15 | dihlsprn 40714 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
| 17 | 10, 12, 16 | syl2anc 583 | . . . 4 β’ (π β (πβ{π}) β ran πΌ) |
| 18 | 17 | adantr 480 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
| 19 | 14, 18 | eqeltrd 2827 | . 2 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) β ran πΌ) |
| 20 | preq2 4733 | . . . . 5 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) | |
| 21 | 20 | fveq2d 6888 | . . . 4 β’ (π = (0gβπ) β (πβ{π, π}) = (πβ{π, (0gβπ)})) |
| 22 | dihprrn.x | . . . . 5 β’ (π β π β π) | |
| 23 | 5, 6, 7, 11, 22 | lsppr0 20937 | . . . 4 β’ (π β (πβ{π, (0gβπ)}) = (πβ{π})) |
| 24 | 21, 23 | sylan9eqr 2788 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) = (πβ{π})) |
| 25 | 8, 9, 5, 7, 15 | dihlsprn 40714 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
| 26 | 10, 22, 25 | syl2anc 583 | . . . 4 β’ (π β (πβ{π}) β ran πΌ) |
| 27 | 26 | adantr 480 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
| 28 | 24, 27 | eqeltrd 2827 | . 2 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) β ran πΌ) |
| 29 | 10 | adantr 480 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β (πΎ β HL β§ π β π»)) |
| 30 | 22 | adantr 480 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β π) |
| 31 | 12 | adantr 480 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β π) |
| 32 | simprl 768 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β (0gβπ)) | |
| 33 | simprr 770 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β (0gβπ)) | |
| 34 | 8, 9, 5, 7, 15, 29, 30, 31, 6, 32, 33 | dihprrnlem2 40808 | . 2 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β (πβ{π, π}) β ran πΌ) |
| 35 | 19, 28, 34 | pm2.61da2ne 3024 | 1 β’ (π β (πβ{π, π}) β ran πΌ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 {csn 4623 {cpr 4625 ran crn 5670 βcfv 6536 Basecbs 17150 0gc0g 17391 LSpanclspn 20815 HLchlt 38732 LHypclh 39367 DVecHcdvh 40461 DIsoHcdih 40611 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-riotaBAD 38335 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-undef 8256 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-0g 17393 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-cntz 19230 df-lsm 19553 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-dvr 20300 df-drng 20586 df-lmod 20705 df-lss 20776 df-lsp 20816 df-lvec 20948 df-lsatoms 38358 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-lplanes 38882 df-lvols 38883 df-lines 38884 df-psubsp 38886 df-pmap 38887 df-padd 39179 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542 df-tgrp 40126 df-tendo 40138 df-edring 40140 df-dveca 40386 df-disoa 40412 df-dvech 40462 df-dib 40522 df-dic 40556 df-dih 40612 df-doch 40731 df-djh 40778 |
| This theorem is referenced by: djhlsmat 40810 lclkrlem2v 40911 lcfrlem23 40948 |
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