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Mirrors > Home > MPE Home > Th. List > ocvlsp | Structured version Visualization version GIF version |
Description: The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
ocvlsp.v | β’ π = (Baseβπ) |
ocvlsp.o | β’ β₯ = (ocvβπ) |
ocvlsp.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
ocvlsp | β’ ((π β PreHil β§ π β π) β ( β₯ β(πβπ)) = ( β₯ βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phllmod 21562 | . . . 4 β’ (π β PreHil β π β LMod) | |
2 | ocvlsp.v | . . . . 5 β’ π = (Baseβπ) | |
3 | ocvlsp.n | . . . . 5 β’ π = (LSpanβπ) | |
4 | 2, 3 | lspssid 20869 | . . . 4 β’ ((π β LMod β§ π β π) β π β (πβπ)) |
5 | 1, 4 | sylan 579 | . . 3 β’ ((π β PreHil β§ π β π) β π β (πβπ)) |
6 | ocvlsp.o | . . . 4 β’ β₯ = (ocvβπ) | |
7 | 6 | ocv2ss 21605 | . . 3 β’ (π β (πβπ) β ( β₯ β(πβπ)) β ( β₯ βπ)) |
8 | 5, 7 | syl 17 | . 2 β’ ((π β PreHil β§ π β π) β ( β₯ β(πβπ)) β ( β₯ βπ)) |
9 | 2, 6 | ocvss 21602 | . . . . 5 β’ ( β₯ βπ) β π |
10 | 9 | a1i 11 | . . . 4 β’ ((π β PreHil β§ π β π) β ( β₯ βπ) β π) |
11 | 2, 6 | ocvocv 21603 | . . . 4 β’ ((π β PreHil β§ ( β₯ βπ) β π) β ( β₯ βπ) β ( β₯ β( β₯ β( β₯ βπ)))) |
12 | 10, 11 | syldan 590 | . . 3 β’ ((π β PreHil β§ π β π) β ( β₯ βπ) β ( β₯ β( β₯ β( β₯ βπ)))) |
13 | 1 | adantr 480 | . . . . 5 β’ ((π β PreHil β§ π β π) β π β LMod) |
14 | eqid 2728 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
15 | 2, 6, 14 | ocvlss 21604 | . . . . . 6 β’ ((π β PreHil β§ ( β₯ βπ) β π) β ( β₯ β( β₯ βπ)) β (LSubSpβπ)) |
16 | 10, 15 | syldan 590 | . . . . 5 β’ ((π β PreHil β§ π β π) β ( β₯ β( β₯ βπ)) β (LSubSpβπ)) |
17 | 2, 6 | ocvocv 21603 | . . . . 5 β’ ((π β PreHil β§ π β π) β π β ( β₯ β( β₯ βπ))) |
18 | 14, 3 | lspssp 20872 | . . . . 5 β’ ((π β LMod β§ ( β₯ β( β₯ βπ)) β (LSubSpβπ) β§ π β ( β₯ β( β₯ βπ))) β (πβπ) β ( β₯ β( β₯ βπ))) |
19 | 13, 16, 17, 18 | syl3anc 1369 | . . . 4 β’ ((π β PreHil β§ π β π) β (πβπ) β ( β₯ β( β₯ βπ))) |
20 | 6 | ocv2ss 21605 | . . . 4 β’ ((πβπ) β ( β₯ β( β₯ βπ)) β ( β₯ β( β₯ β( β₯ βπ))) β ( β₯ β(πβπ))) |
21 | 19, 20 | syl 17 | . . 3 β’ ((π β PreHil β§ π β π) β ( β₯ β( β₯ β( β₯ βπ))) β ( β₯ β(πβπ))) |
22 | 12, 21 | sstrd 3990 | . 2 β’ ((π β PreHil β§ π β π) β ( β₯ βπ) β ( β₯ β(πβπ))) |
23 | 8, 22 | eqssd 3997 | 1 β’ ((π β PreHil β§ π β π) β ( β₯ β(πβπ)) = ( β₯ βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3947 βcfv 6548 Basecbs 17180 LModclmod 20743 LSubSpclss 20815 LSpanclspn 20855 PreHilcphl 21556 ocvcocv 21592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-grp 18893 df-minusg 18894 df-ghm 19168 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-rhm 20411 df-staf 20725 df-srng 20726 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lmhm 20907 df-lvec 20988 df-sra 21058 df-rgmod 21059 df-phl 21558 df-ocv 21595 |
This theorem is referenced by: ocvz 21610 obselocv 21662 obslbs 21664 |
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