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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 2725 | . 2 β’ ((πΊ β LVec β§ π β π) β (BaseβπΊ) = (BaseβπΊ)) | |
2 | tnglvec.t | . . . 4 β’ π = (πΊ toNrmGrp π) | |
3 | eqid 2724 | . . . 4 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | 2, 3 | tngbas 24495 | . . 3 β’ (π β π β (BaseβπΊ) = (Baseβπ)) |
5 | 4 | adantl 481 | . 2 β’ ((πΊ β LVec β§ π β π) β (BaseβπΊ) = (Baseβπ)) |
6 | ssidd 3998 | . 2 β’ ((πΊ β LVec β§ π β π) β (BaseβπΊ) β (BaseβπΊ)) | |
7 | eqid 2724 | . . . . 5 β’ (+gβπΊ) = (+gβπΊ) | |
8 | 2, 7 | tngplusg 24497 | . . . 4 β’ (π β π β (+gβπΊ) = (+gβπ)) |
9 | 8 | adantl 481 | . . 3 β’ ((πΊ β LVec β§ π β π) β (+gβπΊ) = (+gβπ)) |
10 | 9 | oveqdr 7430 | . 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 20950 | . . . 4 β’ (πΊ β LVec β πΊ β LMod) | |
12 | eqid 2724 | . . . . . 6 β’ (ScalarβπΊ) = (ScalarβπΊ) | |
13 | eqid 2724 | . . . . . 6 β’ ( Β·π βπΊ) = ( Β·π βπΊ) | |
14 | eqid 2724 | . . . . . 6 β’ (Baseβ(ScalarβπΊ)) = (Baseβ(ScalarβπΊ)) | |
15 | 3, 12, 13, 14 | lmodvscl 20720 | . . . . 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 1117 | . . . 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 579 | . . 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 712 | . 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 24504 | . . . 4 β’ (π β π β ( Β·π βπΊ) = ( Β·π βπ)) |
20 | 19 | adantl 481 | . . 3 β’ ((πΊ β LVec β§ π β π) β ( Β·π βπΊ) = ( Β·π βπ)) |
21 | 20 | oveqdr 7430 | . 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 2724 | . 2 β’ (Scalarβπ) = (Scalarβπ) | |
23 | eqidd 2725 | . 2 β’ ((πΊ β LVec β§ π β π) β (Baseβ(ScalarβπΊ)) = (Baseβ(ScalarβπΊ))) | |
24 | 2, 12 | tngsca 24502 | . . . 4 β’ (π β π β (ScalarβπΊ) = (Scalarβπ)) |
25 | 24 | adantl 481 | . . 3 β’ ((πΊ β LVec β§ π β π) β (ScalarβπΊ) = (Scalarβπ)) |
26 | 25 | fveq2d 6886 | . 2 β’ ((πΊ β LVec β§ π β π) β (Baseβ(ScalarβπΊ)) = (Baseβ(Scalarβπ))) |
27 | 25 | fveq2d 6886 | . . 3 β’ ((πΊ β LVec β§ π β π) β (+gβ(ScalarβπΊ)) = (+gβ(Scalarβπ))) |
28 | 27 | oveqdr 7430 | . 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 482 | . 2 β’ ((πΊ β LVec β§ π β π) β πΊ β LVec) | |
30 | 2 | tnglvec 33205 | . . 3 β’ (π β π β (πΊ β LVec β π β LVec)) |
31 | 30 | biimpac 478 | . 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 33201 | 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 395 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 Basecbs 17149 +gcplusg 17202 Scalarcsca 17205 Β·π cvsca 17206 LModclmod 20702 LVecclvec 20946 toNrmGrp ctng 24431 dimcldim 33191 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-reg 9584 ax-inf2 9633 ax-ac2 10455 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-rpss 7707 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-oi 9502 df-r1 9756 df-rank 9757 df-dju 9893 df-card 9931 df-acn 9934 df-ac 10108 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-xnn0 12544 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ocomp 17223 df-ds 17224 df-0g 17392 df-mre 17535 df-mrc 17536 df-mri 17537 df-acs 17538 df-proset 18256 df-drs 18257 df-poset 18274 df-ipo 18489 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lbs 20919 df-lvec 20947 df-tng 24437 df-dim 33192 |
This theorem is referenced by: rrxdim 33207 |
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