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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 20452). (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 4783 | . 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 20452 | . 2 β’ ((π β LMod β§ {π, π} β π) β (πβ{π, π}) β π) |
9 | 1, 4, 8 | syl2anc 585 | 1 β’ (π β (πβ{π, π}) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3911 {cpr 4589 βcfv 6497 Basecbs 17088 LModclmod 20336 LSubSpclss 20407 LSpanclspn 20447 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 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-op 4594 df-uni 4867 df-int 4909 df-iun 4957 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-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: lspprid1 20473 lspprvacl 20475 lsmelpr 20567 lspexch 20606 lspindpi 20609 lsppratlem4 20627 lsatfixedN 37517 dvh3dim2 39957 dvh3dim3N 39958 lclkrlem2v 40037 lcfrlem23 40074 lcfrlem25 40076 mapdindp 40180 baerlem3lem1 40216 baerlem5alem1 40217 baerlem5blem1 40218 baerlem5amN 40225 baerlem5bmN 40226 baerlem5abmN 40227 mapdh6aN 40244 mapdh6b0N 40245 mapdh6iN 40253 lspindp5 40279 mapdh8ab 40286 mapdh8ad 40288 mapdh8e 40293 mapdh9a 40298 mapdh9aOLDN 40299 hdmap1l6a 40318 hdmap1l6b0N 40319 hdmap1l6i 40327 hdmap1eulemOLDN 40332 hdmapval0 40342 hdmapval3lemN 40346 hdmap10lem 40348 hdmap11lem1 40350 hdmap11lem2 40351 hdmap14lem11 40387 |
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