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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 21519 | . . . 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 20830 | . . . 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 21562 | . . 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 21559 | . . . . 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 21560 | . . . 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 2726 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
15 | 2, 6, 14 | ocvlss 21561 | . . . . . 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 21560 | . . . . 5 β’ ((π β PreHil β§ π β π) β π β ( β₯ β( β₯ βπ))) |
18 | 14, 3 | lspssp 20833 | . . . . 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 1368 | . . . 4 β’ ((π β PreHil β§ π β π) β (πβπ) β ( β₯ β( β₯ βπ))) |
20 | 6 | ocv2ss 21562 | . . . 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 3987 | . 2 β’ ((π β PreHil β§ π β π) β ( β₯ βπ) β ( β₯ β(πβπ))) |
23 | 8, 22 | eqssd 3994 | 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 1533 β wcel 2098 β wss 3943 βcfv 6536 Basecbs 17151 LModclmod 20704 LSubSpclss 20776 LSpanclspn 20816 PreHilcphl 21513 ocvcocv 21549 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-minusg 18865 df-ghm 19137 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20234 df-rhm 20372 df-staf 20686 df-srng 20687 df-lmod 20706 df-lss 20777 df-lsp 20817 df-lmhm 20868 df-lvec 20949 df-sra 21019 df-rgmod 21020 df-phl 21515 df-ocv 21552 |
This theorem is referenced by: ocvz 21567 obselocv 21619 obslbs 21621 |
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