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 4741 | . . . . . 6 β’ {π, π} = {π, π} | |
| 2 | preq2 4743 | . . . . . 6 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) | |
| 3 | 1, 2 | eqtrid 2780 | . . . . 5 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) |
| 4 | 3 | fveq2d 6906 | . . . 4 β’ (π = (0gβπ) β (πβ{π, π}) = (πβ{π, (0gβπ)})) |
| 5 | dihprrn.v | . . . . 5 β’ π = (Baseβπ) | |
| 6 | eqid 2728 | . . . . 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 40615 | . . . . 5 β’ (π β π β LMod) |
| 12 | dihprrn.y | . . . . 5 β’ (π β π β π) | |
| 13 | 5, 6, 7, 11, 12 | lsppr0 20984 | . . . 4 β’ (π β (πβ{π, (0gβπ)}) = (πβ{π})) |
| 14 | 4, 13 | sylan9eqr 2790 | . . 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 40836 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
| 17 | 10, 12, 16 | syl2anc 582 | . . . 4 β’ (π β (πβ{π}) β ran πΌ) |
| 18 | 17 | adantr 479 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
| 19 | 14, 18 | eqeltrd 2829 | . 2 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) β ran πΌ) |
| 20 | preq2 4743 | . . . . 5 β’ (π = (0gβπ) β {π, π} = {π, (0gβπ)}) | |
| 21 | 20 | fveq2d 6906 | . . . 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 20984 | . . . 4 β’ (π β (πβ{π, (0gβπ)}) = (πβ{π})) |
| 24 | 21, 23 | sylan9eqr 2790 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) = (πβ{π})) |
| 25 | 8, 9, 5, 7, 15 | dihlsprn 40836 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πβ{π}) β ran πΌ) |
| 26 | 10, 22, 25 | syl2anc 582 | . . . 4 β’ (π β (πβ{π}) β ran πΌ) |
| 27 | 26 | adantr 479 | . . 3 β’ ((π β§ π = (0gβπ)) β (πβ{π}) β ran πΌ) |
| 28 | 24, 27 | eqeltrd 2829 | . 2 β’ ((π β§ π = (0gβπ)) β (πβ{π, π}) β ran πΌ) |
| 29 | 10 | adantr 479 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β (πΎ β HL β§ π β π»)) |
| 30 | 22 | adantr 479 | . . 3 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β π β π) |
| 31 | 12 | adantr 479 | . . 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 40930 | . 2 β’ ((π β§ (π β (0gβπ) β§ π β (0gβπ))) β (πβ{π, π}) β ran πΌ) |
| 35 | 19, 28, 34 | pm2.61da2ne 3027 | 1 β’ (π β (πβ{π, π}) β ran πΌ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 {csn 4632 {cpr 4634 ran crn 5683 βcfv 6553 Basecbs 17187 0gc0g 17428 LSpanclspn 20862 HLchlt 38854 LHypclh 39489 DVecHcdvh 40583 DIsoHcdih 40733 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38457 |
| 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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-undef 8285 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-0g 17430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lsatoms 38480 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-tgrp 40248 df-tendo 40260 df-edring 40262 df-dveca 40508 df-disoa 40534 df-dvech 40584 df-dib 40644 df-dic 40678 df-dih 40734 df-doch 40853 df-djh 40900 |
| This theorem is referenced by: djhlsmat 40932 lclkrlem2v 41033 lcfrlem23 41070 |
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