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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihoml4 | Structured version Visualization version GIF version |
Description: Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 39458 analog.) (Contributed by NM, 15-Jan-2015.) |
Ref | Expression |
---|---|
dihoml4.h | β’ π» = (LHypβπΎ) |
dihoml4.u | β’ π = ((DVecHβπΎ)βπ) |
dihoml4.s | β’ π = (LSubSpβπ) |
dihoml4.o | β’ β₯ = ((ocHβπΎ)βπ) |
dihoml4.k | β’ (π β (πΎ β HL β§ π β π»)) |
dihoml4.x | β’ (π β π β π) |
dihoml4.y | β’ (π β π β π) |
dihoml4.c | β’ (π β ( β₯ β( β₯ βπ)) = π) |
dihoml4.l | β’ (π β π β π) |
Ref | Expression |
---|---|
dihoml4 | β’ (π β (( β₯ β(( β₯ βπ) β© π)) β© π) = ( β₯ β( β₯ βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihoml4.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
2 | dihoml4.x | . . . . . . . 8 β’ (π β π β π) | |
3 | eqid 2728 | . . . . . . . . 9 β’ (Baseβπ) = (Baseβπ) | |
4 | dihoml4.s | . . . . . . . . 9 β’ π = (LSubSpβπ) | |
5 | 3, 4 | lssss 20827 | . . . . . . . 8 β’ (π β π β π β (Baseβπ)) |
6 | 2, 5 | syl 17 | . . . . . . 7 β’ (π β π β (Baseβπ)) |
7 | dihoml4.h | . . . . . . . 8 β’ π» = (LHypβπΎ) | |
8 | eqid 2728 | . . . . . . . 8 β’ ((DIsoHβπΎ)βπ) = ((DIsoHβπΎ)βπ) | |
9 | dihoml4.u | . . . . . . . 8 β’ π = ((DVecHβπΎ)βπ) | |
10 | dihoml4.o | . . . . . . . 8 β’ β₯ = ((ocHβπΎ)βπ) | |
11 | 7, 8, 9, 3, 10 | dochcl 40858 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ π β (Baseβπ)) β ( β₯ βπ) β ran ((DIsoHβπΎ)βπ)) |
12 | 1, 6, 11 | syl2anc 582 | . . . . . 6 β’ (π β ( β₯ βπ) β ran ((DIsoHβπΎ)βπ)) |
13 | 7, 8, 10 | dochoc 40872 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ ( β₯ βπ) β ran ((DIsoHβπΎ)βπ)) β ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ)) |
14 | 1, 12, 13 | syl2anc 582 | . . . . 5 β’ (π β ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ)) |
15 | 14 | ineq1d 4213 | . . . 4 β’ (π β (( β₯ β( β₯ β( β₯ βπ))) β© π) = (( β₯ βπ) β© π)) |
16 | 15 | fveq2d 6906 | . . 3 β’ (π β ( β₯ β(( β₯ β( β₯ β( β₯ βπ))) β© π)) = ( β₯ β(( β₯ βπ) β© π))) |
17 | 16 | ineq1d 4213 | . 2 β’ (π β (( β₯ β(( β₯ β( β₯ β( β₯ βπ))) β© π)) β© π) = (( β₯ β(( β₯ βπ) β© π)) β© π)) |
18 | 7, 9, 3, 10 | dochssv 40860 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β (Baseβπ)) β ( β₯ βπ) β (Baseβπ)) |
19 | 1, 6, 18 | syl2anc 582 | . . . 4 β’ (π β ( β₯ βπ) β (Baseβπ)) |
20 | 7, 8, 9, 3, 10 | dochcl 40858 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ ( β₯ βπ) β (Baseβπ)) β ( β₯ β( β₯ βπ)) β ran ((DIsoHβπΎ)βπ)) |
21 | 1, 19, 20 | syl2anc 582 | . . 3 β’ (π β ( β₯ β( β₯ βπ)) β ran ((DIsoHβπΎ)βπ)) |
22 | dihoml4.c | . . . 4 β’ (π β ( β₯ β( β₯ βπ)) = π) | |
23 | dihoml4.y | . . . . . 6 β’ (π β π β π) | |
24 | 3, 4 | lssss 20827 | . . . . . 6 β’ (π β π β π β (Baseβπ)) |
25 | 23, 24 | syl 17 | . . . . 5 β’ (π β π β (Baseβπ)) |
26 | 7, 8, 9, 3, 10, 1, 25 | dochoccl 40874 | . . . 4 β’ (π β (π β ran ((DIsoHβπΎ)βπ) β ( β₯ β( β₯ βπ)) = π)) |
27 | 22, 26 | mpbird 256 | . . 3 β’ (π β π β ran ((DIsoHβπΎ)βπ)) |
28 | dihoml4.l | . . . . . 6 β’ (π β π β π) | |
29 | 7, 9, 3, 10 | dochss 40870 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ π β (Baseβπ) β§ π β π) β ( β₯ βπ) β ( β₯ βπ)) |
30 | 1, 25, 28, 29 | syl3anc 1368 | . . . . 5 β’ (π β ( β₯ βπ) β ( β₯ βπ)) |
31 | 7, 9, 3, 10 | dochss 40870 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ ( β₯ βπ) β (Baseβπ) β§ ( β₯ βπ) β ( β₯ βπ)) β ( β₯ β( β₯ βπ)) β ( β₯ β( β₯ βπ))) |
32 | 1, 19, 30, 31 | syl3anc 1368 | . . . 4 β’ (π β ( β₯ β( β₯ βπ)) β ( β₯ β( β₯ βπ))) |
33 | 32, 22 | sseqtrd 4022 | . . 3 β’ (π β ( β₯ β( β₯ βπ)) β π) |
34 | 7, 8, 10, 1, 21, 27, 33 | dihoml4c 40881 | . 2 β’ (π β (( β₯ β(( β₯ β( β₯ β( β₯ βπ))) β© π)) β© π) = ( β₯ β( β₯ βπ))) |
35 | 17, 34 | eqtr3d 2770 | 1 β’ (π β (( β₯ β(( β₯ βπ) β© π)) β© π) = ( β₯ β( β₯ βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β© cin 3948 β wss 3949 ran crn 5683 βcfv 6553 Basecbs 17187 LSubSpclss 20822 HLchlt 38854 LHypclh 39489 DVecHcdvh 40583 DIsoHcdih 40733 ocHcoch 40852 |
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-tendo 40260 df-edring 40262 df-disoa 40534 df-dvech 40584 df-dib 40644 df-dic 40678 df-dih 40734 df-doch 40853 |
This theorem is referenced by: dochexmidlem6 40970 |
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