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| Mirrors > Home > MPE Home > Th. List > lspprcl | Structured version Visualization version GIF version | ||
| Description: The span of a pair is a subspace (frequently used special case of lspcl 20867). (Contributed by NM, 11-Apr-2015.) |
| Ref | Expression |
|---|---|
| lspval.v | β’ π = (Baseβπ) |
| lspval.s | β’ π = (LSubSpβπ) |
| lspval.n | β’ π = (LSpanβπ) |
| lspprcl.w | β’ (π β π β LMod) |
| lspprcl.x | β’ (π β π β π) |
| lspprcl.y | β’ (π β π β π) |
| Ref | Expression |
|---|---|
| lspprcl | β’ (π β (πβ{π, π}) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspprcl.w | . 2 β’ (π β π β LMod) | |
| 2 | lspprcl.x | . . 3 β’ (π β π β π) | |
| 3 | lspprcl.y | . . 3 β’ (π β π β π) | |
| 4 | 2, 3 | prssd 4830 | . 2 β’ (π β {π, π} β π) |
| 5 | lspval.v | . . 3 β’ π = (Baseβπ) | |
| 6 | lspval.s | . . 3 β’ π = (LSubSpβπ) | |
| 7 | lspval.n | . . 3 β’ π = (LSpanβπ) | |
| 8 | 5, 6, 7 | lspcl 20867 | . 2 β’ ((π β LMod β§ {π, π} β π) β (πβ{π, π}) β π) |
| 9 | 1, 4, 8 | syl2anc 582 | 1 β’ (π β (πβ{π, π}) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3949 {cpr 4634 βcfv 6553 Basecbs 17187 LModclmod 20750 LSubSpclss 20822 LSpanclspn 20862 |
| 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 |
| 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-op 4639 df-uni 4913 df-int 4954 df-iun 5002 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-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 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-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mgp 20082 df-ur 20129 df-ring 20182 df-lmod 20752 df-lss 20823 df-lsp 20863 |
| This theorem is referenced by: lspprid1 20888 lspprvacl 20890 lsmelpr 20983 lspexch 21024 lspindpi 21027 lsppratlem4 21045 lsatfixedN 38513 dvh3dim2 40953 dvh3dim3N 40954 lclkrlem2v 41033 lcfrlem23 41070 lcfrlem25 41072 mapdindp 41176 baerlem3lem1 41212 baerlem5alem1 41213 baerlem5blem1 41214 baerlem5amN 41221 baerlem5bmN 41222 baerlem5abmN 41223 mapdh6aN 41240 mapdh6b0N 41241 mapdh6iN 41249 lspindp5 41275 mapdh8ab 41282 mapdh8ad 41284 mapdh8e 41289 mapdh9a 41294 mapdh9aOLDN 41295 hdmap1l6a 41314 hdmap1l6b0N 41315 hdmap1l6i 41323 hdmap1eulemOLDN 41328 hdmapval0 41338 hdmapval3lemN 41342 hdmap10lem 41344 hdmap11lem1 41346 hdmap11lem2 41347 hdmap14lem11 41383 |
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