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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8ab | Structured version Visualization version GIF version |
Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-May-2015.) |
Ref | Expression |
---|---|
mapdh8a.h | β’ π» = (LHypβπΎ) |
mapdh8a.u | β’ π = ((DVecHβπΎ)βπ) |
mapdh8a.v | β’ π = (Baseβπ) |
mapdh8a.s | β’ β = (-gβπ) |
mapdh8a.o | β’ 0 = (0gβπ) |
mapdh8a.n | β’ π = (LSpanβπ) |
mapdh8a.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdh8a.d | β’ π· = (BaseβπΆ) |
mapdh8a.r | β’ π = (-gβπΆ) |
mapdh8a.q | β’ π = (0gβπΆ) |
mapdh8a.j | β’ π½ = (LSpanβπΆ) |
mapdh8a.m | β’ π = ((mapdβπΎ)βπ) |
mapdh8a.i | β’ πΌ = (π₯ β V β¦ if((2nd βπ₯) = 0 , π, (β©β β π· ((πβ(πβ{(2nd βπ₯)})) = (π½β{β}) β§ (πβ(πβ{((1st β(1st βπ₯)) β (2nd βπ₯))})) = (π½β{((2nd β(1st βπ₯))π β)}))))) |
mapdh8a.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdh8ab.f | β’ (π β πΉ β π·) |
mapdh8ab.mn | β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) |
mapdh8ab.eg | β’ (π β (πΌββ¨π, πΉ, πβ©) = πΊ) |
mapdh8ab.ee | β’ (π β (πΌββ¨π, πΉ, πβ©) = πΈ) |
mapdh8ab.x | β’ (π β π β (π β { 0 })) |
mapdh8ab.y | β’ (π β π β (π β { 0 })) |
mapdh8ab.z | β’ (π β π β (π β { 0 })) |
mapdh8ab.t | β’ (π β π β (π β { 0 })) |
mapdh8ab.yz | β’ (π β (πβ{π}) β (πβ{π})) |
mapdh8ab.xn | β’ (π β Β¬ π β (πβ{π, π})) |
mapdh8ab.yn | β’ (π β (πβ{π}) = (πβ{π})) |
Ref | Expression |
---|---|
mapdh8ab | β’ (π β (πΌββ¨π, πΊ, πβ©) = (πΌββ¨π, πΈ, πβ©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh8a.h | . 2 β’ π» = (LHypβπΎ) | |
2 | mapdh8a.u | . 2 β’ π = ((DVecHβπΎ)βπ) | |
3 | mapdh8a.v | . 2 β’ π = (Baseβπ) | |
4 | mapdh8a.s | . 2 β’ β = (-gβπ) | |
5 | mapdh8a.o | . 2 β’ 0 = (0gβπ) | |
6 | mapdh8a.n | . 2 β’ π = (LSpanβπ) | |
7 | mapdh8a.c | . 2 β’ πΆ = ((LCDualβπΎ)βπ) | |
8 | mapdh8a.d | . 2 β’ π· = (BaseβπΆ) | |
9 | mapdh8a.r | . 2 β’ π = (-gβπΆ) | |
10 | mapdh8a.q | . 2 β’ π = (0gβπΆ) | |
11 | mapdh8a.j | . 2 β’ π½ = (LSpanβπΆ) | |
12 | mapdh8a.m | . 2 β’ π = ((mapdβπΎ)βπ) | |
13 | mapdh8a.i | . 2 β’ πΌ = (π₯ β V β¦ if((2nd βπ₯) = 0 , π, (β©β β π· ((πβ(πβ{(2nd βπ₯)})) = (π½β{β}) β§ (πβ(πβ{((1st β(1st βπ₯)) β (2nd βπ₯))})) = (π½β{((2nd β(1st βπ₯))π β)}))))) | |
14 | mapdh8a.k | . 2 β’ (π β (πΎ β HL β§ π β π»)) | |
15 | mapdh8ab.f | . 2 β’ (π β πΉ β π·) | |
16 | mapdh8ab.mn | . 2 β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) | |
17 | mapdh8ab.eg | . 2 β’ (π β (πΌββ¨π, πΉ, πβ©) = πΊ) | |
18 | mapdh8ab.ee | . 2 β’ (π β (πΌββ¨π, πΉ, πβ©) = πΈ) | |
19 | mapdh8ab.x | . 2 β’ (π β π β (π β { 0 })) | |
20 | mapdh8ab.y | . 2 β’ (π β π β (π β { 0 })) | |
21 | mapdh8ab.z | . 2 β’ (π β π β (π β { 0 })) | |
22 | 1, 2, 14 | dvhlvec 39601 | . . . . . 6 β’ (π β π β LVec) |
23 | 19 | eldifad 3927 | . . . . . 6 β’ (π β π β π) |
24 | 20 | eldifad 3927 | . . . . . 6 β’ (π β π β π) |
25 | 21 | eldifad 3927 | . . . . . 6 β’ (π β π β π) |
26 | mapdh8ab.xn | . . . . . 6 β’ (π β Β¬ π β (πβ{π, π})) | |
27 | 3, 6, 22, 23, 24, 25, 26 | lspindpi 20609 | . . . . 5 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
28 | 27 | simprd 497 | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) |
29 | 28 | necomd 3000 | . . 3 β’ (π β (πβ{π}) β (πβ{π})) |
30 | mapdh8ab.yn | . . 3 β’ (π β (πβ{π}) = (πβ{π})) | |
31 | 29, 30 | neeqtrd 3014 | . 2 β’ (π β (πβ{π}) β (πβ{π})) |
32 | mapdh8ab.t | . 2 β’ (π β π β (π β { 0 })) | |
33 | 30 | sseq1d 3980 | . . . . 5 β’ (π β ((πβ{π}) β (πβ{π, π}) β (πβ{π}) β (πβ{π, π}))) |
34 | eqid 2737 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
35 | 1, 2, 14 | dvhlmod 39602 | . . . . . 6 β’ (π β π β LMod) |
36 | 3, 34, 6, 35, 24, 25 | lspprcl 20455 | . . . . . 6 β’ (π β (πβ{π, π}) β (LSubSpβπ)) |
37 | 3, 34, 6, 35, 36, 23 | lspsnel5 20472 | . . . . 5 β’ (π β (π β (πβ{π, π}) β (πβ{π}) β (πβ{π, π}))) |
38 | 32 | eldifad 3927 | . . . . . 6 β’ (π β π β π) |
39 | 3, 34, 6, 35, 36, 38 | lspsnel5 20472 | . . . . 5 β’ (π β (π β (πβ{π, π}) β (πβ{π}) β (πβ{π, π}))) |
40 | 33, 37, 39 | 3bitr4d 311 | . . . 4 β’ (π β (π β (πβ{π, π}) β π β (πβ{π, π}))) |
41 | 26, 40 | mtbid 324 | . . 3 β’ (π β Β¬ π β (πβ{π, π})) |
42 | 22 | adantr 482 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β LVec) |
43 | 20 | adantr 482 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β (π β { 0 })) |
44 | 38 | adantr 482 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β π) |
45 | 25 | adantr 482 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β π) |
46 | mapdh8ab.yz | . . . . 5 β’ (π β (πβ{π}) β (πβ{π})) | |
47 | 46 | adantr 482 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β (πβ{π}) β (πβ{π})) |
48 | simpr 486 | . . . . 5 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) | |
49 | prcom 4698 | . . . . . 6 β’ {π, π} = {π, π} | |
50 | 49 | fveq2i 6850 | . . . . 5 β’ (πβ{π, π}) = (πβ{π, π}) |
51 | 48, 50 | eleqtrdi 2848 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) |
52 | 3, 5, 6, 42, 43, 44, 45, 47, 51 | lspexch 20606 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) |
53 | 41, 52 | mtand 815 | . 2 β’ (π β Β¬ π β (πβ{π, π})) |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 31, 32, 53, 26 | mapdh8aa 40268 | 1 β’ (π β (πΌββ¨π, πΊ, πβ©) = (πΌββ¨π, πΈ, πβ©)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 Vcvv 3448 β cdif 3912 β wss 3915 ifcif 4491 {csn 4591 {cpr 4593 β¨cotp 4599 β¦ cmpt 5193 βcfv 6501 β©crio 7317 (class class class)co 7362 1st c1st 7924 2nd c2nd 7925 Basecbs 17090 0gc0g 17328 -gcsg 18757 LSubSpclss 20408 LSpanclspn 20448 LVecclvec 20579 HLchlt 37841 LHypclh 38476 DVecHcdvh 39570 LCDualclcd 40078 mapdcmpd 40116 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-riotaBAD 37444 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-undef 8209 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-0g 17330 df-mre 17473 df-mrc 17474 df-acs 17476 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-cntz 19104 df-oppg 19131 df-lsm 19425 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-oppr 20056 df-dvdsr 20077 df-unit 20078 df-invr 20108 df-dvr 20119 df-drng 20201 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lvec 20580 df-lsatoms 37467 df-lshyp 37468 df-lcv 37510 df-lfl 37549 df-lkr 37577 df-ldual 37615 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-llines 37990 df-lplanes 37991 df-lvols 37992 df-lines 37993 df-psubsp 37995 df-pmap 37996 df-padd 38288 df-lhyp 38480 df-laut 38481 df-ldil 38596 df-ltrn 38597 df-trl 38651 df-tgrp 39235 df-tendo 39247 df-edring 39249 df-dveca 39495 df-disoa 39521 df-dvech 39571 df-dib 39631 df-dic 39665 df-dih 39721 df-doch 39840 df-djh 39887 df-lcdual 40079 df-mapd 40117 |
This theorem is referenced by: mapdh8ac 40270 |
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