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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tngdim | Structured version Visualization version GIF version |
Description: Dimension of a left vector space augmented with a norm. (Contributed by Thierry Arnoux, 20-May-2023.) |
Ref | Expression |
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tnglvec.t | β’ π = (πΊ toNrmGrp π) |
Ref | Expression |
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tngdim | β’ ((πΊ β LVec β§ π β π) β (dimβπΊ) = (dimβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2734 | . 2 β’ ((πΊ β LVec β§ π β π) β (BaseβπΊ) = (BaseβπΊ)) | |
2 | tnglvec.t | . . . 4 β’ π = (πΊ toNrmGrp π) | |
3 | eqid 2733 | . . . 4 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | 2, 3 | tngbas 24021 | . . 3 β’ (π β π β (BaseβπΊ) = (Baseβπ)) |
5 | 4 | adantl 483 | . 2 β’ ((πΊ β LVec β§ π β π) β (BaseβπΊ) = (Baseβπ)) |
6 | ssidd 3971 | . 2 β’ ((πΊ β LVec β§ π β π) β (BaseβπΊ) β (BaseβπΊ)) | |
7 | eqid 2733 | . . . . 5 β’ (+gβπΊ) = (+gβπΊ) | |
8 | 2, 7 | tngplusg 24023 | . . . 4 β’ (π β π β (+gβπΊ) = (+gβπ)) |
9 | 8 | adantl 483 | . . 3 β’ ((πΊ β LVec β§ π β π) β (+gβπΊ) = (+gβπ)) |
10 | 9 | oveqdr 7389 | . 2 β’ (((πΊ β LVec β§ π β π) β§ (π₯ β (BaseβπΊ) β§ π¦ β (BaseβπΊ))) β (π₯(+gβπΊ)π¦) = (π₯(+gβπ)π¦)) |
11 | lveclmod 20611 | . . . 4 β’ (πΊ β LVec β πΊ β LMod) | |
12 | eqid 2733 | . . . . . 6 β’ (ScalarβπΊ) = (ScalarβπΊ) | |
13 | eqid 2733 | . . . . . 6 β’ ( Β·π βπΊ) = ( Β·π βπΊ) | |
14 | eqid 2733 | . . . . . 6 β’ (Baseβ(ScalarβπΊ)) = (Baseβ(ScalarβπΊ)) | |
15 | 3, 12, 13, 14 | lmodvscl 20383 | . . . . 5 β’ ((πΊ β LMod β§ π₯ β (Baseβ(ScalarβπΊ)) β§ π¦ β (BaseβπΊ)) β (π₯( Β·π βπΊ)π¦) β (BaseβπΊ)) |
16 | 15 | 3expb 1121 | . . . 4 β’ ((πΊ β LMod β§ (π₯ β (Baseβ(ScalarβπΊ)) β§ π¦ β (BaseβπΊ))) β (π₯( Β·π βπΊ)π¦) β (BaseβπΊ)) |
17 | 11, 16 | sylan 581 | . . 3 β’ ((πΊ β LVec β§ (π₯ β (Baseβ(ScalarβπΊ)) β§ π¦ β (BaseβπΊ))) β (π₯( Β·π βπΊ)π¦) β (BaseβπΊ)) |
18 | 17 | adantlr 714 | . 2 β’ (((πΊ β LVec β§ π β π) β§ (π₯ β (Baseβ(ScalarβπΊ)) β§ π¦ β (BaseβπΊ))) β (π₯( Β·π βπΊ)π¦) β (BaseβπΊ)) |
19 | 2, 13 | tngvsca 24030 | . . . 4 β’ (π β π β ( Β·π βπΊ) = ( Β·π βπ)) |
20 | 19 | adantl 483 | . . 3 β’ ((πΊ β LVec β§ π β π) β ( Β·π βπΊ) = ( Β·π βπ)) |
21 | 20 | oveqdr 7389 | . 2 β’ (((πΊ β LVec β§ π β π) β§ (π₯ β (Baseβ(ScalarβπΊ)) β§ π¦ β (BaseβπΊ))) β (π₯( Β·π βπΊ)π¦) = (π₯( Β·π βπ)π¦)) |
22 | eqid 2733 | . 2 β’ (Scalarβπ) = (Scalarβπ) | |
23 | eqidd 2734 | . 2 β’ ((πΊ β LVec β§ π β π) β (Baseβ(ScalarβπΊ)) = (Baseβ(ScalarβπΊ))) | |
24 | 2, 12 | tngsca 24028 | . . . 4 β’ (π β π β (ScalarβπΊ) = (Scalarβπ)) |
25 | 24 | adantl 483 | . . 3 β’ ((πΊ β LVec β§ π β π) β (ScalarβπΊ) = (Scalarβπ)) |
26 | 25 | fveq2d 6850 | . 2 β’ ((πΊ β LVec β§ π β π) β (Baseβ(ScalarβπΊ)) = (Baseβ(Scalarβπ))) |
27 | 25 | fveq2d 6850 | . . 3 β’ ((πΊ β LVec β§ π β π) β (+gβ(ScalarβπΊ)) = (+gβ(Scalarβπ))) |
28 | 27 | oveqdr 7389 | . 2 β’ (((πΊ β LVec β§ π β π) β§ (π₯ β (Baseβ(ScalarβπΊ)) β§ π¦ β (Baseβ(ScalarβπΊ)))) β (π₯(+gβ(ScalarβπΊ))π¦) = (π₯(+gβ(Scalarβπ))π¦)) |
29 | simpl 484 | . 2 β’ ((πΊ β LVec β§ π β π) β πΊ β LVec) | |
30 | 2 | tnglvec 32371 | . . 3 β’ (π β π β (πΊ β LVec β π β LVec)) |
31 | 30 | biimpac 480 | . 2 β’ ((πΊ β LVec β§ π β π) β π β LVec) |
32 | 1, 5, 6, 10, 18, 21, 12, 22, 23, 26, 28, 29, 31 | dimpropd 32368 | 1 β’ ((πΊ β LVec β§ π β π) β (dimβπΊ) = (dimβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 Scalarcsca 17144 Β·π cvsca 17145 LModclmod 20365 LVecclvec 20607 toNrmGrp ctng 23957 dimcldim 32360 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-reg 9536 ax-inf2 9585 ax-ac2 10407 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-rpss 7664 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-oi 9454 df-r1 9708 df-rank 9709 df-dju 9845 df-card 9883 df-acn 9886 df-ac 10060 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ocomp 17162 df-ds 17163 df-0g 17331 df-mre 17474 df-mrc 17475 df-mri 17476 df-acs 17477 df-proset 18192 df-drs 18193 df-poset 18210 df-ipo 18425 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-drng 20221 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lbs 20580 df-lvec 20608 df-tng 23963 df-dim 32361 |
This theorem is referenced by: rrxdim 32373 |
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