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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem1a | Structured version Visualization version GIF version |
Description: Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.) |
Ref | Expression |
---|---|
hdmap14lem1a.h | β’ π» = (LHypβπΎ) |
hdmap14lem1a.u | β’ π = ((DVecHβπΎ)βπ) |
hdmap14lem1a.v | β’ π = (Baseβπ) |
hdmap14lem1a.t | β’ Β· = ( Β·π βπ) |
hdmap14lem1a.r | β’ π = (Scalarβπ) |
hdmap14lem1a.b | β’ π΅ = (Baseβπ ) |
hdmap14lem1a.c | β’ πΆ = ((LCDualβπΎ)βπ) |
hdmap14lem2a.e | β’ β = ( Β·π βπΆ) |
hdmap14lem1a.l | β’ πΏ = (LSpanβπΆ) |
hdmap14lem2a.p | β’ π = (ScalarβπΆ) |
hdmap14lem2a.a | β’ π΄ = (Baseβπ) |
hdmap14lem1a.s | β’ π = ((HDMapβπΎ)βπ) |
hdmap14lem1a.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmap14lem3a.x | β’ (π β π β π) |
hdmap14lem1a.f | β’ (π β πΉ β π΅) |
hdmap14lem1a.z | β’ 0 = (0gβπ ) |
hdmap14lem1a.fn | β’ (π β πΉ β 0 ) |
Ref | Expression |
---|---|
hdmap14lem1a | β’ (π β (πΏβ{(πβπ)}) = (πΏβ{(πβ(πΉ Β· π))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem1a.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | hdmap14lem1a.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
3 | hdmap14lem1a.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | dvhlvec 40470 | . . . 4 β’ (π β π β LVec) |
5 | hdmap14lem1a.f | . . . 4 β’ (π β πΉ β π΅) | |
6 | hdmap14lem1a.fn | . . . 4 β’ (π β πΉ β 0 ) | |
7 | hdmap14lem3a.x | . . . 4 β’ (π β π β π) | |
8 | hdmap14lem1a.v | . . . . 5 β’ π = (Baseβπ) | |
9 | hdmap14lem1a.r | . . . . 5 β’ π = (Scalarβπ) | |
10 | hdmap14lem1a.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
11 | hdmap14lem1a.b | . . . . 5 β’ π΅ = (Baseβπ ) | |
12 | hdmap14lem1a.z | . . . . 5 β’ 0 = (0gβπ ) | |
13 | eqid 2724 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
14 | 8, 9, 10, 11, 12, 13 | lspsnvs 20955 | . . . 4 β’ ((π β LVec β§ (πΉ β π΅ β§ πΉ β 0 ) β§ π β π) β ((LSpanβπ)β{(πΉ Β· π)}) = ((LSpanβπ)β{π})) |
15 | 4, 5, 6, 7, 14 | syl121anc 1372 | . . 3 β’ (π β ((LSpanβπ)β{(πΉ Β· π)}) = ((LSpanβπ)β{π})) |
16 | 15 | fveq2d 6885 | . 2 β’ (π β (((mapdβπΎ)βπ)β((LSpanβπ)β{(πΉ Β· π)})) = (((mapdβπΎ)βπ)β((LSpanβπ)β{π}))) |
17 | hdmap14lem1a.c | . . 3 β’ πΆ = ((LCDualβπΎ)βπ) | |
18 | hdmap14lem1a.l | . . 3 β’ πΏ = (LSpanβπΆ) | |
19 | eqid 2724 | . . 3 β’ ((mapdβπΎ)βπ) = ((mapdβπΎ)βπ) | |
20 | hdmap14lem1a.s | . . 3 β’ π = ((HDMapβπΎ)βπ) | |
21 | 1, 2, 3 | dvhlmod 40471 | . . . 4 β’ (π β π β LMod) |
22 | 8, 9, 10, 11 | lmodvscl 20714 | . . . 4 β’ ((π β LMod β§ πΉ β π΅ β§ π β π) β (πΉ Β· π) β π) |
23 | 21, 5, 7, 22 | syl3anc 1368 | . . 3 β’ (π β (πΉ Β· π) β π) |
24 | 1, 2, 8, 13, 17, 18, 19, 20, 3, 23 | hdmap10 41201 | . 2 β’ (π β (((mapdβπΎ)βπ)β((LSpanβπ)β{(πΉ Β· π)})) = (πΏβ{(πβ(πΉ Β· π))})) |
25 | 1, 2, 8, 13, 17, 18, 19, 20, 3, 7 | hdmap10 41201 | . 2 β’ (π β (((mapdβπΎ)βπ)β((LSpanβπ)β{π})) = (πΏβ{(πβπ)})) |
26 | 16, 24, 25 | 3eqtr3rd 2773 | 1 β’ (π β (πΏβ{(πβπ)}) = (πΏβ{(πβ(πΉ Β· π))})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 {csn 4620 βcfv 6533 (class class class)co 7401 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 0gc0g 17384 LModclmod 20696 LSpanclspn 20808 LVecclvec 20940 HLchlt 38710 LHypclh 39345 DVecHcdvh 40439 LCDualclcd 40947 mapdcmpd 40985 HDMapchdma 41153 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 38313 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17386 df-mre 17529 df-mrc 17530 df-acs 17532 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-cntz 19223 df-oppg 19252 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20579 df-lmod 20698 df-lss 20769 df-lsp 20809 df-lvec 20941 df-lsatoms 38336 df-lshyp 38337 df-lcv 38379 df-lfl 38418 df-lkr 38446 df-ldual 38484 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-llines 38859 df-lplanes 38860 df-lvols 38861 df-lines 38862 df-psubsp 38864 df-pmap 38865 df-padd 39157 df-lhyp 39349 df-laut 39350 df-ldil 39465 df-ltrn 39466 df-trl 39520 df-tgrp 40104 df-tendo 40116 df-edring 40118 df-dveca 40364 df-disoa 40390 df-dvech 40440 df-dib 40500 df-dic 40534 df-dih 40590 df-doch 40709 df-djh 40756 df-lcdual 40948 df-mapd 40986 df-hvmap 41118 df-hdmap1 41154 df-hdmap 41155 |
This theorem is referenced by: hdmap14lem2a 41228 hdmap14lem1 41229 |
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