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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem3N | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 15, T β P. Our (β‘πβ(πΏβ{((πβπ’) β π )})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmaprnlem1.h | β’ π» = (LHypβπΎ) |
hdmaprnlem1.u | β’ π = ((DVecHβπΎ)βπ) |
hdmaprnlem1.v | β’ π = (Baseβπ) |
hdmaprnlem1.n | β’ π = (LSpanβπ) |
hdmaprnlem1.c | β’ πΆ = ((LCDualβπΎ)βπ) |
hdmaprnlem1.l | β’ πΏ = (LSpanβπΆ) |
hdmaprnlem1.m | β’ π = ((mapdβπΎ)βπ) |
hdmaprnlem1.s | β’ π = ((HDMapβπΎ)βπ) |
hdmaprnlem1.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmaprnlem1.se | β’ (π β π β (π· β {π})) |
hdmaprnlem1.ve | β’ (π β π£ β π) |
hdmaprnlem1.e | β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) |
hdmaprnlem1.ue | β’ (π β π’ β π) |
hdmaprnlem1.un | β’ (π β Β¬ π’ β (πβ{π£})) |
hdmaprnlem1.d | β’ π· = (BaseβπΆ) |
hdmaprnlem1.q | β’ π = (0gβπΆ) |
hdmaprnlem1.o | β’ 0 = (0gβπ) |
hdmaprnlem1.a | β’ β = (+gβπΆ) |
Ref | Expression |
---|---|
hdmaprnlem3N | β’ (π β (πβ{π£}) β (β‘πβ(πΏβ{((πβπ’) β π )}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmaprnlem1.d | . . . . 5 β’ π· = (BaseβπΆ) | |
2 | hdmaprnlem1.l | . . . . 5 β’ πΏ = (LSpanβπΆ) | |
3 | hdmaprnlem1.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
4 | hdmaprnlem1.c | . . . . . 6 β’ πΆ = ((LCDualβπΎ)βπ) | |
5 | hdmaprnlem1.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
6 | 3, 4, 5 | lcdlmod 40989 | . . . . 5 β’ (π β πΆ β LMod) |
7 | hdmaprnlem1.u | . . . . . . 7 β’ π = ((DVecHβπΎ)βπ) | |
8 | hdmaprnlem1.v | . . . . . . 7 β’ π = (Baseβπ) | |
9 | hdmaprnlem1.s | . . . . . . 7 β’ π = ((HDMapβπΎ)βπ) | |
10 | hdmaprnlem1.ue | . . . . . . 7 β’ (π β π’ β π) | |
11 | 3, 7, 8, 4, 1, 9, 5, 10 | hdmapcl 41227 | . . . . . 6 β’ (π β (πβπ’) β π·) |
12 | hdmaprnlem1.se | . . . . . . 7 β’ (π β π β (π· β {π})) | |
13 | 12 | eldifad 3956 | . . . . . 6 β’ (π β π β π·) |
14 | hdmaprnlem1.a | . . . . . . 7 β’ β = (+gβπΆ) | |
15 | 1, 14 | lmodvacl 20740 | . . . . . 6 β’ ((πΆ β LMod β§ (πβπ’) β π· β§ π β π·) β ((πβπ’) β π ) β π·) |
16 | 6, 11, 13, 15 | syl3anc 1369 | . . . . 5 β’ (π β ((πβπ’) β π ) β π·) |
17 | eqid 2727 | . . . . . 6 β’ (LSubSpβπΆ) = (LSubSpβπΆ) | |
18 | 1, 17, 2 | lspsncl 20843 | . . . . . . 7 β’ ((πΆ β LMod β§ π β π·) β (πΏβ{π }) β (LSubSpβπΆ)) |
19 | 6, 13, 18 | syl2anc 583 | . . . . . 6 β’ (π β (πΏβ{π }) β (LSubSpβπΆ)) |
20 | 1, 2 | lspsnid 20859 | . . . . . . 7 β’ ((πΆ β LMod β§ π β π·) β π β (πΏβ{π })) |
21 | 6, 13, 20 | syl2anc 583 | . . . . . 6 β’ (π β π β (πΏβ{π })) |
22 | hdmaprnlem1.q | . . . . . . 7 β’ π = (0gβπΆ) | |
23 | 3, 4, 5 | lcdlvec 40988 | . . . . . . 7 β’ (π β πΆ β LVec) |
24 | hdmaprnlem1.o | . . . . . . . 8 β’ 0 = (0gβπ) | |
25 | eqid 2727 | . . . . . . . . 9 β’ (LSubSpβπ) = (LSubSpβπ) | |
26 | 3, 7, 5 | dvhlmod 40507 | . . . . . . . . 9 β’ (π β π β LMod) |
27 | hdmaprnlem1.ve | . . . . . . . . . 10 β’ (π β π£ β π) | |
28 | hdmaprnlem1.n | . . . . . . . . . . 11 β’ π = (LSpanβπ) | |
29 | 8, 25, 28 | lspsncl 20843 | . . . . . . . . . 10 β’ ((π β LMod β§ π£ β π) β (πβ{π£}) β (LSubSpβπ)) |
30 | 26, 27, 29 | syl2anc 583 | . . . . . . . . 9 β’ (π β (πβ{π£}) β (LSubSpβπ)) |
31 | hdmaprnlem1.un | . . . . . . . . 9 β’ (π β Β¬ π’ β (πβ{π£})) | |
32 | 24, 25, 26, 30, 10, 31 | lssneln0 20819 | . . . . . . . 8 β’ (π β π’ β (π β { 0 })) |
33 | 3, 7, 8, 24, 4, 22, 1, 9, 5, 32 | hdmapnzcl 41242 | . . . . . . 7 β’ (π β (πβπ’) β (π· β {π})) |
34 | hdmaprnlem1.m | . . . . . . . 8 β’ π = ((mapdβπΎ)βπ) | |
35 | hdmaprnlem1.e | . . . . . . . 8 β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) | |
36 | 3, 7, 8, 28, 4, 2, 34, 9, 5, 12, 27, 35, 10, 31 | hdmaprnlem1N 41246 | . . . . . . 7 β’ (π β (πΏβ{(πβπ’)}) β (πΏβ{π })) |
37 | 1, 22, 2, 23, 33, 13, 36 | lspsnne1 20987 | . . . . . 6 β’ (π β Β¬ (πβπ’) β (πΏβ{π })) |
38 | 1, 14, 17, 6, 19, 21, 11, 37 | lssvancl2 20812 | . . . . 5 β’ (π β Β¬ ((πβπ’) β π ) β (πΏβ{π })) |
39 | 1, 2, 6, 16, 13, 38 | lspsnne2 20988 | . . . 4 β’ (π β (πΏβ{((πβπ’) β π )}) β (πΏβ{π })) |
40 | 39 | necomd 2991 | . . 3 β’ (π β (πΏβ{π }) β (πΏβ{((πβπ’) β π )})) |
41 | 1, 17, 2 | lspsncl 20843 | . . . . . 6 β’ ((πΆ β LMod β§ ((πβπ’) β π ) β π·) β (πΏβ{((πβπ’) β π )}) β (LSubSpβπΆ)) |
42 | 6, 16, 41 | syl2anc 583 | . . . . 5 β’ (π β (πΏβ{((πβπ’) β π )}) β (LSubSpβπΆ)) |
43 | 3, 34, 4, 17, 5 | mapdrn2 41048 | . . . . 5 β’ (π β ran π = (LSubSpβπΆ)) |
44 | 42, 43 | eleqtrrd 2831 | . . . 4 β’ (π β (πΏβ{((πβπ’) β π )}) β ran π) |
45 | 3, 34, 5, 44 | mapdcnvid2 41054 | . . 3 β’ (π β (πβ(β‘πβ(πΏβ{((πβπ’) β π )}))) = (πΏβ{((πβπ’) β π )})) |
46 | 40, 35, 45 | 3netr4d 3013 | . 2 β’ (π β (πβ(πβ{π£})) β (πβ(β‘πβ(πΏβ{((πβπ’) β π )})))) |
47 | 3, 34, 7, 25, 5, 44 | mapdcnvcl 41049 | . . . 4 β’ (π β (β‘πβ(πΏβ{((πβπ’) β π )})) β (LSubSpβπ)) |
48 | 3, 7, 25, 34, 5, 30, 47 | mapd11 41036 | . . 3 β’ (π β ((πβ(πβ{π£})) = (πβ(β‘πβ(πΏβ{((πβπ’) β π )}))) β (πβ{π£}) = (β‘πβ(πΏβ{((πβπ’) β π )})))) |
49 | 48 | necon3bid 2980 | . 2 β’ (π β ((πβ(πβ{π£})) β (πβ(β‘πβ(πΏβ{((πβπ’) β π )}))) β (πβ{π£}) β (β‘πβ(πΏβ{((πβπ’) β π )})))) |
50 | 46, 49 | mpbid 231 | 1 β’ (π β (πβ{π£}) β (β‘πβ(πΏβ{((πβπ’) β π )}))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2935 β cdif 3941 {csn 4624 β‘ccnv 5671 ran crn 5673 βcfv 6542 (class class class)co 7414 Basecbs 17165 +gcplusg 17218 0gc0g 17406 LModclmod 20725 LSubSpclss 20797 LSpanclspn 20837 HLchlt 38746 LHypclh 39381 DVecHcdvh 40475 LCDualclcd 40983 mapdcmpd 41021 HDMapchdma 41189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 38349 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-tpos 8223 df-undef 8270 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17408 df-mre 17551 df-mrc 17552 df-acs 17554 df-proset 18272 df-poset 18290 df-plt 18307 df-lub 18323 df-glb 18324 df-join 18325 df-meet 18326 df-p0 18402 df-p1 18403 df-lat 18409 df-clat 18476 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19062 df-cntz 19252 df-oppg 19281 df-lsm 19575 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-oppr 20255 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-drng 20608 df-lmod 20727 df-lss 20798 df-lsp 20838 df-lvec 20970 df-lsatoms 38372 df-lshyp 38373 df-lcv 38415 df-lfl 38454 df-lkr 38482 df-ldual 38520 df-oposet 38572 df-ol 38574 df-oml 38575 df-covers 38662 df-ats 38663 df-atl 38694 df-cvlat 38718 df-hlat 38747 df-llines 38895 df-lplanes 38896 df-lvols 38897 df-lines 38898 df-psubsp 38900 df-pmap 38901 df-padd 39193 df-lhyp 39385 df-laut 39386 df-ldil 39501 df-ltrn 39502 df-trl 39556 df-tgrp 40140 df-tendo 40152 df-edring 40154 df-dveca 40400 df-disoa 40426 df-dvech 40476 df-dib 40536 df-dic 40570 df-dih 40626 df-doch 40745 df-djh 40792 df-lcdual 40984 df-mapd 41022 df-hvmap 41154 df-hdmap1 41190 df-hdmap 41191 |
This theorem is referenced by: hdmaprnlem9N 41254 hdmaprnlem3eN 41255 |
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