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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdheq2biN | Structured version Visualization version GIF version |
Description: Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 39785 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdh.q | β’ π = (0gβπΆ) |
mapdh.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)}))))) |
mapdh.h | β’ π» = (LHypβπΎ) |
mapdh.m | β’ π = ((mapdβπΎ)βπ) |
mapdh.u | β’ π = ((DVecHβπΎ)βπ) |
mapdh.v | β’ π = (Baseβπ) |
mapdh.s | β’ β = (-gβπ) |
mapdhc.o | β’ 0 = (0gβπ) |
mapdh.n | β’ π = (LSpanβπ) |
mapdh.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdh.d | β’ π· = (BaseβπΆ) |
mapdh.r | β’ π = (-gβπΆ) |
mapdh.j | β’ π½ = (LSpanβπΆ) |
mapdh.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdhc.f | β’ (π β πΉ β π·) |
mapdh.mn | β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) |
mapdhcl.x | β’ (π β π β (π β { 0 })) |
mapdhe2.y | β’ (π β π β (π β { 0 })) |
mapdhe2.g | β’ (π β πΊ β π·) |
mapdh.ne3 | β’ (π β (πβ{π}) β (πβ{π})) |
mapdh.my | β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) |
Ref | Expression |
---|---|
mapdheq2biN | β’ (π β ((πΌββ¨π, πΉ, πβ©) = πΊ β (πΌββ¨π, πΊ, πβ©) = πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.q | . . 3 β’ π = (0gβπΆ) | |
2 | mapdh.i | . . 3 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)}))))) | |
3 | mapdh.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | mapdh.m | . . 3 β’ π = ((mapdβπΎ)βπ) | |
5 | mapdh.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
6 | mapdh.v | . . 3 β’ π = (Baseβπ) | |
7 | mapdh.s | . . 3 β’ β = (-gβπ) | |
8 | mapdhc.o | . . 3 β’ 0 = (0gβπ) | |
9 | mapdh.n | . . 3 β’ π = (LSpanβπ) | |
10 | mapdh.c | . . 3 β’ πΆ = ((LCDualβπΎ)βπ) | |
11 | mapdh.d | . . 3 β’ π· = (BaseβπΆ) | |
12 | mapdh.r | . . 3 β’ π = (-gβπΆ) | |
13 | mapdh.j | . . 3 β’ π½ = (LSpanβπΆ) | |
14 | mapdh.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
15 | mapdhc.f | . . 3 β’ (π β πΉ β π·) | |
16 | mapdh.mn | . . 3 β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) | |
17 | mapdhcl.x | . . 3 β’ (π β π β (π β { 0 })) | |
18 | mapdhe2.y | . . 3 β’ (π β π β (π β { 0 })) | |
19 | mapdhe2.g | . . 3 β’ (π β πΊ β π·) | |
20 | mapdh.ne3 | . . 3 β’ (π β (πβ{π}) β (πβ{π})) | |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 | mapdheq2 39785 | . 2 β’ (π β ((πΌββ¨π, πΉ, πβ©) = πΊ β (πΌββ¨π, πΊ, πβ©) = πΉ)) |
22 | mapdh.my | . . 3 β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) | |
23 | 20 | necomd 2997 | . . 3 β’ (π β (πβ{π}) β (πβ{π})) |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 22, 18, 17, 15, 23 | mapdheq2 39785 | . 2 β’ (π β ((πΌββ¨π, πΊ, πβ©) = πΉ β (πΌββ¨π, πΉ, πβ©) = πΊ)) |
25 | 21, 24 | impbid 211 | 1 β’ (π β ((πΌββ¨π, πΉ, πβ©) = πΊ β (πΌββ¨π, πΊ, πβ©) = πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1539 β wcel 2104 β wne 2941 Vcvv 3437 β cdif 3889 ifcif 4465 {csn 4565 β¨cotp 4573 β¦ cmpt 5164 βcfv 6458 β©crio 7263 (class class class)co 7307 1st c1st 7861 2nd c2nd 7862 Basecbs 16957 0gc0g 17195 -gcsg 18624 LSpanclspn 20278 HLchlt 37406 LHypclh 38040 DVecHcdvh 39134 LCDualclcd 39642 mapdcmpd 39680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-riotaBAD 37009 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-ot 4574 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-tpos 8073 df-undef 8120 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-sca 17023 df-vsca 17024 df-0g 17197 df-mre 17340 df-mrc 17341 df-acs 17343 df-proset 18058 df-poset 18076 df-plt 18093 df-lub 18109 df-glb 18110 df-join 18111 df-meet 18112 df-p0 18188 df-p1 18189 df-lat 18195 df-clat 18262 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-grp 18625 df-minusg 18626 df-sbg 18627 df-subg 18797 df-cntz 18968 df-oppg 18995 df-lsm 19286 df-cmn 19433 df-abl 19434 df-mgp 19766 df-ur 19783 df-ring 19830 df-oppr 19907 df-dvdsr 19928 df-unit 19929 df-invr 19959 df-dvr 19970 df-drng 20038 df-lmod 20170 df-lss 20239 df-lsp 20279 df-lvec 20410 df-lsatoms 37032 df-lshyp 37033 df-lcv 37075 df-lfl 37114 df-lkr 37142 df-ldual 37180 df-oposet 37232 df-ol 37234 df-oml 37235 df-covers 37322 df-ats 37323 df-atl 37354 df-cvlat 37378 df-hlat 37407 df-llines 37554 df-lplanes 37555 df-lvols 37556 df-lines 37557 df-psubsp 37559 df-pmap 37560 df-padd 37852 df-lhyp 38044 df-laut 38045 df-ldil 38160 df-ltrn 38161 df-trl 38215 df-tgrp 38799 df-tendo 38811 df-edring 38813 df-dveca 39059 df-disoa 39085 df-dvech 39135 df-dib 39195 df-dic 39229 df-dih 39285 df-doch 39404 df-djh 39451 df-lcdual 39643 df-mapd 39681 |
This theorem is referenced by: (None) |
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