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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvbase | Structured version Visualization version GIF version |
Description: The vectors (vector base set) of the constructed full vector space H are all translations (for a fiducial co-atom π). (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Ref | Expression |
---|---|
dvhvbase.h | β’ π» = (LHypβπΎ) |
dvhvbase.t | β’ π = ((LTrnβπΎ)βπ) |
dvhvbase.e | β’ πΈ = ((TEndoβπΎ)βπ) |
dvhvbase.u | β’ π = ((DVecHβπΎ)βπ) |
dvhvbase.v | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
dvhvbase | β’ ((πΎ β π β§ π β π») β π = (π Γ πΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvhvbase.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | dvhvbase.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
3 | dvhvbase.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
4 | eqid 2737 | . . . 4 β’ ((EDRingβπΎ)βπ) = ((EDRingβπΎ)βπ) | |
5 | dvhvbase.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | dvhset 39547 | . . 3 β’ ((πΎ β π β§ π β π») β π = ({β¨(Baseβndx), (π Γ πΈ)β©, β¨(+gβndx), (π β (π Γ πΈ), π β (π Γ πΈ) β¦ β¨((1st βπ) β (1st βπ)), (β β π β¦ (((2nd βπ)ββ) β ((2nd βπ)ββ)))β©)β©, β¨(Scalarβndx), ((EDRingβπΎ)βπ)β©} βͺ {β¨( Β·π βndx), (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)β©})) |
7 | 6 | fveq2d 6847 | . 2 β’ ((πΎ β π β§ π β π») β (Baseβπ) = (Baseβ({β¨(Baseβndx), (π Γ πΈ)β©, β¨(+gβndx), (π β (π Γ πΈ), π β (π Γ πΈ) β¦ β¨((1st βπ) β (1st βπ)), (β β π β¦ (((2nd βπ)ββ) β ((2nd βπ)ββ)))β©)β©, β¨(Scalarβndx), ((EDRingβπΎ)βπ)β©} βͺ {β¨( Β·π βndx), (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)β©}))) |
8 | dvhvbase.v | . 2 β’ π = (Baseβπ) | |
9 | 2 | fvexi 6857 | . . . 4 β’ π β V |
10 | 3 | fvexi 6857 | . . . 4 β’ πΈ β V |
11 | 9, 10 | xpex 7688 | . . 3 β’ (π Γ πΈ) β V |
12 | eqid 2737 | . . . 4 β’ ({β¨(Baseβndx), (π Γ πΈ)β©, β¨(+gβndx), (π β (π Γ πΈ), π β (π Γ πΈ) β¦ β¨((1st βπ) β (1st βπ)), (β β π β¦ (((2nd βπ)ββ) β ((2nd βπ)ββ)))β©)β©, β¨(Scalarβndx), ((EDRingβπΎ)βπ)β©} βͺ {β¨( Β·π βndx), (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)β©}) = ({β¨(Baseβndx), (π Γ πΈ)β©, β¨(+gβndx), (π β (π Γ πΈ), π β (π Γ πΈ) β¦ β¨((1st βπ) β (1st βπ)), (β β π β¦ (((2nd βπ)ββ) β ((2nd βπ)ββ)))β©)β©, β¨(Scalarβndx), ((EDRingβπΎ)βπ)β©} βͺ {β¨( Β·π βndx), (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)β©}) | |
13 | 12 | lmodbase 17208 | . . 3 β’ ((π Γ πΈ) β V β (π Γ πΈ) = (Baseβ({β¨(Baseβndx), (π Γ πΈ)β©, β¨(+gβndx), (π β (π Γ πΈ), π β (π Γ πΈ) β¦ β¨((1st βπ) β (1st βπ)), (β β π β¦ (((2nd βπ)ββ) β ((2nd βπ)ββ)))β©)β©, β¨(Scalarβndx), ((EDRingβπΎ)βπ)β©} βͺ {β¨( Β·π βndx), (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)β©}))) |
14 | 11, 13 | ax-mp 5 | . 2 β’ (π Γ πΈ) = (Baseβ({β¨(Baseβndx), (π Γ πΈ)β©, β¨(+gβndx), (π β (π Γ πΈ), π β (π Γ πΈ) β¦ β¨((1st βπ) β (1st βπ)), (β β π β¦ (((2nd βπ)ββ) β ((2nd βπ)ββ)))β©)β©, β¨(Scalarβndx), ((EDRingβπΎ)βπ)β©} βͺ {β¨( Β·π βndx), (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)β©})) |
15 | 7, 8, 14 | 3eqtr4g 2802 | 1 β’ ((πΎ β π β§ π β π») β π = (π Γ πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3446 βͺ cun 3909 {csn 4587 {ctp 4591 β¨cop 4593 β¦ cmpt 5189 Γ cxp 5632 β ccom 5638 βcfv 6497 β cmpo 7360 1st c1st 7920 2nd c2nd 7921 ndxcnx 17066 Basecbs 17084 +gcplusg 17134 Scalarcsca 17137 Β·π cvsca 17138 LHypclh 38450 LTrncltrn 38567 TEndoctendo 39218 EDRingcedring 39219 DVecHcdvh 39544 |
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 |
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-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-iun 4957 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-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 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-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-sca 17150 df-vsca 17151 df-dvech 39545 |
This theorem is referenced by: dvhelvbasei 39554 dvhgrp 39573 dvhlveclem 39574 dvhopellsm 39583 dibss 39635 diblss 39636 dicssdvh 39652 dicelval1stN 39654 dicelval2nd 39655 dicvaddcl 39656 dicvscacl 39657 diclss 39659 dihssxp 39718 dihvalrel 39745 dih1 39752 dih1dimatlem 39795 |
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