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Mathbox for Norm Megill |
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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 40614 | . . . . . 6 β’ (π β π β LVec) |
23 | 19 | eldifad 3961 | . . . . . 6 β’ (π β π β π) |
24 | 20 | eldifad 3961 | . . . . . 6 β’ (π β π β π) |
25 | 21 | eldifad 3961 | . . . . . 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 21027 | . . . . 5 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
28 | 27 | simprd 494 | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) |
29 | 28 | necomd 2993 | . . 3 β’ (π β (πβ{π}) β (πβ{π})) |
30 | mapdh8ab.yn | . . 3 β’ (π β (πβ{π}) = (πβ{π})) | |
31 | 29, 30 | neeqtrd 3007 | . 2 β’ (π β (πβ{π}) β (πβ{π})) |
32 | mapdh8ab.t | . 2 β’ (π β π β (π β { 0 })) | |
33 | 30 | sseq1d 4013 | . . . . 5 β’ (π β ((πβ{π}) β (πβ{π, π}) β (πβ{π}) β (πβ{π, π}))) |
34 | eqid 2728 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
35 | 1, 2, 14 | dvhlmod 40615 | . . . . . 6 β’ (π β π β LMod) |
36 | 3, 34, 6, 35, 24, 25 | lspprcl 20869 | . . . . . 6 β’ (π β (πβ{π, π}) β (LSubSpβπ)) |
37 | 3, 34, 6, 35, 36, 23 | lspsnel5 20886 | . . . . 5 β’ (π β (π β (πβ{π, π}) β (πβ{π}) β (πβ{π, π}))) |
38 | 32 | eldifad 3961 | . . . . . 6 β’ (π β π β π) |
39 | 3, 34, 6, 35, 36, 38 | lspsnel5 20886 | . . . . 5 β’ (π β (π β (πβ{π, π}) β (πβ{π}) β (πβ{π, π}))) |
40 | 33, 37, 39 | 3bitr4d 310 | . . . 4 β’ (π β (π β (πβ{π, π}) β π β (πβ{π, π}))) |
41 | 26, 40 | mtbid 323 | . . 3 β’ (π β Β¬ π β (πβ{π, π})) |
42 | 22 | adantr 479 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β LVec) |
43 | 20 | adantr 479 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β (π β { 0 })) |
44 | 38 | adantr 479 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β π) |
45 | 25 | adantr 479 | . . . 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 479 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β (πβ{π}) β (πβ{π})) |
48 | simpr 483 | . . . . 5 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) | |
49 | prcom 4741 | . . . . . 6 β’ {π, π} = {π, π} | |
50 | 49 | fveq2i 6905 | . . . . 5 β’ (πβ{π, π}) = (πβ{π, π}) |
51 | 48, 50 | eleqtrdi 2839 | . . . 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 21024 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) |
53 | 41, 52 | mtand 814 | . 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 41281 | 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 394 = wceq 1533 β wcel 2098 β wne 2937 Vcvv 3473 β cdif 3946 β wss 3949 ifcif 4532 {csn 4632 {cpr 4634 β¨cotp 4640 β¦ cmpt 5235 βcfv 6553 β©crio 7381 (class class class)co 7426 1st c1st 7997 2nd c2nd 7998 Basecbs 17187 0gc0g 17428 -gcsg 18899 LSubSpclss 20822 LSpanclspn 20862 LVecclvec 20994 HLchlt 38854 LHypclh 39489 DVecHcdvh 40583 LCDualclcd 41091 mapdcmpd 41129 |
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-ot 4641 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-of 7691 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-mre 17573 df-mrc 17574 df-acs 17576 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-oppg 19304 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-lshyp 38481 df-lcv 38523 df-lfl 38562 df-lkr 38590 df-ldual 38628 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-tgrp 40248 df-tendo 40260 df-edring 40262 df-dveca 40508 df-disoa 40534 df-dvech 40584 df-dib 40644 df-dic 40678 df-dih 40734 df-doch 40853 df-djh 40900 df-lcdual 41092 df-mapd 41130 |
This theorem is referenced by: mapdh8ac 41283 |
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