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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem4N | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 19. (T* =) (Ft)* = Gs. (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) |
hdmaprnlem1.t2 | β’ (π β π‘ β ((πβ{π£}) β { 0 })) |
Ref | Expression |
---|---|
hdmaprnlem4N | β’ (π β (πβ(πβ{π‘})) = (πΏβ{π })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
2 | hdmaprnlem1.n | . . . . 5 β’ π = (LSpanβπ) | |
3 | hdmaprnlem1.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
4 | hdmaprnlem1.u | . . . . . 6 β’ π = ((DVecHβπΎ)βπ) | |
5 | hdmaprnlem1.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
6 | 3, 4, 5 | dvhlmod 39576 | . . . . 5 β’ (π β π β LMod) |
7 | hdmaprnlem1.ve | . . . . . 6 β’ (π β π£ β π) | |
8 | hdmaprnlem1.v | . . . . . . 7 β’ π = (Baseβπ) | |
9 | 8, 1, 2 | lspsncl 20441 | . . . . . 6 β’ ((π β LMod β§ π£ β π) β (πβ{π£}) β (LSubSpβπ)) |
10 | 6, 7, 9 | syl2anc 585 | . . . . 5 β’ (π β (πβ{π£}) β (LSubSpβπ)) |
11 | hdmaprnlem1.t2 | . . . . . 6 β’ (π β π‘ β ((πβ{π£}) β { 0 })) | |
12 | 11 | eldifad 3923 | . . . . 5 β’ (π β π‘ β (πβ{π£})) |
13 | 1, 2, 6, 10, 12 | lspsnel5a 20460 | . . . 4 β’ (π β (πβ{π‘}) β (πβ{π£})) |
14 | hdmaprnlem1.o | . . . . 5 β’ 0 = (0gβπ) | |
15 | 3, 4, 5 | dvhlvec 39575 | . . . . 5 β’ (π β π β LVec) |
16 | 8, 1 | lss1 20402 | . . . . . . . . 9 β’ (π β LMod β π β (LSubSpβπ)) |
17 | 6, 16 | syl 17 | . . . . . . . 8 β’ (π β π β (LSubSpβπ)) |
18 | 1, 2, 6, 17, 7 | lspsnel5a 20460 | . . . . . . 7 β’ (π β (πβ{π£}) β π) |
19 | 18 | ssdifd 4101 | . . . . . 6 β’ (π β ((πβ{π£}) β { 0 }) β (π β { 0 })) |
20 | 19, 11 | sseldd 3946 | . . . . 5 β’ (π β π‘ β (π β { 0 })) |
21 | 8, 14, 2, 15, 20, 7 | lspsncmp 20580 | . . . 4 β’ (π β ((πβ{π‘}) β (πβ{π£}) β (πβ{π‘}) = (πβ{π£}))) |
22 | 13, 21 | mpbid 231 | . . 3 β’ (π β (πβ{π‘}) = (πβ{π£})) |
23 | 22 | fveq2d 6847 | . 2 β’ (π β (πβ(πβ{π‘})) = (πβ(πβ{π£}))) |
24 | hdmaprnlem1.e | . 2 β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) | |
25 | 23, 24 | eqtrd 2777 | 1 β’ (π β (πβ(πβ{π‘})) = (πΏβ{π })) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β cdif 3908 β wss 3911 {csn 4587 βcfv 6497 Basecbs 17084 +gcplusg 17134 0gc0g 17322 LModclmod 20325 LSubSpclss 20395 LSpanclspn 20435 HLchlt 37815 LHypclh 38450 DVecHcdvh 39544 LCDualclcd 40052 mapdcmpd 40090 HDMapchdma 40258 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-riotaBAD 37418 |
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 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-undef 8205 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-0g 17324 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-sbg 18754 df-mgp 19898 df-ur 19915 df-ring 19967 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-drng 20188 df-lmod 20327 df-lss 20396 df-lsp 20436 df-lvec 20567 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-tendo 39221 df-edring 39223 df-dvech 39545 |
This theorem is referenced by: hdmaprnlem8N 40322 hdmaprnlem9N 40323 |
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