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 4736 | . . . . . 6 β’ {π, π} = {π, π} | |
| 2 | preq2 4738 | . . . . . 6 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) | |
| 3 | 1, 2 | eqtrid 2784 | . . . . 5 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) |
| 4 | 3 | fveq2d 6895 | . . . 4 β’ (π = (0gβπ) β (πβ{π, π}) = (πβ{π, (0gβπ)})) |
| 5 | dihprrn.v | . . . . 5 β’ π = (Baseβπ) | |
| 6 | eqid 2732 | . . . . 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 39976 | . . . . 5 β’ (π β π β LMod) |
| 12 | dihprrn.y | . . . . 5 β’ (π β π β π) | |
| 13 | 5, 6, 7, 11, 12 | lsppr0 20702 | . . . 4 β’ (π β (πβ{π, (0gβπ)}) = (πβ{π})) |
| 14 | 4, 13 | sylan9eqr 2794 | . . 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 40197 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
| 17 | 10, 12, 16 | syl2anc 584 | . . . 4 β’ (π β (πβ{π}) β ran πΌ) |
| 18 | 17 | adantr 481 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
| 19 | 14, 18 | eqeltrd 2833 | . 2 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) β ran πΌ) |
| 20 | preq2 4738 | . . . . 5 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) | |
| 21 | 20 | fveq2d 6895 | . . . 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 20702 | . . . 4 β’ (π β (πβ{π, (0gβπ)}) = (πβ{π})) |
| 24 | 21, 23 | sylan9eqr 2794 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) = (πβ{π})) |
| 25 | 8, 9, 5, 7, 15 | dihlsprn 40197 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
| 26 | 10, 22, 25 | syl2anc 584 | . . . 4 β’ (π β (πβ{π}) β ran πΌ) |
| 27 | 26 | adantr 481 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
| 28 | 24, 27 | eqeltrd 2833 | . 2 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) β ran πΌ) |
| 29 | 10 | adantr 481 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β (πΎ β HL β§ π β π»)) |
| 30 | 22 | adantr 481 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β π) |
| 31 | 12 | adantr 481 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β π) |
| 32 | simprl 769 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β (0gβπ)) | |
| 33 | simprr 771 | . . 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 40291 | . 2 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β (πβ{π, π}) β ran πΌ) |
| 35 | 19, 28, 34 | pm2.61da2ne 3030 | 1 β’ (π β (πβ{π, π}) β ran πΌ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 {csn 4628 {cpr 4630 ran crn 5677 βcfv 6543 Basecbs 17143 0gc0g 17384 LSpanclspn 20581 HLchlt 38215 LHypclh 38850 DVecHcdvh 39944 DIsoHcdih 40094 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 37818 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-undef 8257 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17386 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-clat 18451 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-subg 19002 df-cntz 19180 df-lsm 19503 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-drng 20358 df-lmod 20472 df-lss 20542 df-lsp 20582 df-lvec 20713 df-lsatoms 37841 df-oposet 38041 df-ol 38043 df-oml 38044 df-covers 38131 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 df-llines 38364 df-lplanes 38365 df-lvols 38366 df-lines 38367 df-psubsp 38369 df-pmap 38370 df-padd 38662 df-lhyp 38854 df-laut 38855 df-ldil 38970 df-ltrn 38971 df-trl 39025 df-tgrp 39609 df-tendo 39621 df-edring 39623 df-dveca 39869 df-disoa 39895 df-dvech 39945 df-dib 40005 df-dic 40039 df-dih 40095 df-doch 40214 df-djh 40261 |
| This theorem is referenced by: djhlsmat 40293 lclkrlem2v 40394 lcfrlem23 40431 |
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