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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 |
---|---|
tnglvec.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
Ref | Expression |
---|---|
tngdim | ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2735 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐺)) | |
2 | tnglvec.t | . . . 4 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
3 | eqid 2734 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | tngbas 24670 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
5 | 4 | adantl 481 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝑇)) |
6 | ssidd 4018 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) ⊆ (Base‘𝐺)) | |
7 | eqid 2734 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | 2, 7 | tngplusg 24672 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
9 | 8 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝑇)) |
10 | 9 | oveqdr 7458 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
11 | lveclmod 21122 | . . . 4 ⊢ (𝐺 ∈ LVec → 𝐺 ∈ LMod) | |
12 | eqid 2734 | . . . . . 6 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺) | |
13 | eqid 2734 | . . . . . 6 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺) | |
14 | eqid 2734 | . . . . . 6 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺)) | |
15 | 3, 12, 13, 14 | lmodvscl 20892 | . . . . 5 ⊢ ((𝐺 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
16 | 15 | 3expb 1119 | . . . 4 ⊢ ((𝐺 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
17 | 11, 16 | sylan 580 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
18 | 17 | adantlr 715 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
19 | 2, 13 | tngvsca 24679 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
20 | 19 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
21 | 20 | oveqdr 7458 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) = (𝑥( ·𝑠 ‘𝑇)𝑦)) |
22 | eqid 2734 | . 2 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
23 | eqidd 2735 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺))) | |
24 | 2, 12 | tngsca 24677 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Scalar‘𝐺) = (Scalar‘𝑇)) |
25 | 24 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝑇)) |
26 | 25 | fveq2d 6910 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑇))) |
27 | 25 | fveq2d 6910 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘(Scalar‘𝐺)) = (+g‘(Scalar‘𝑇))) |
28 | 27 | oveqdr 7458 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘(Scalar‘𝐺)))) → (𝑥(+g‘(Scalar‘𝐺))𝑦) = (𝑥(+g‘(Scalar‘𝑇))𝑦)) |
29 | simpl 482 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ LVec) | |
30 | 2 | tnglvec 33639 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec)) |
31 | 30 | biimpac 478 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝑇 ∈ LVec) |
32 | 1, 5, 6, 10, 18, 21, 12, 22, 23, 26, 28, 29, 31 | dimpropd 33635 | 1 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 LModclmod 20874 LVecclvec 21118 toNrmGrp ctng 24606 dimcldim 33625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-reg 9629 ax-inf2 9678 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-rpss 7741 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-oi 9547 df-r1 9801 df-rank 9802 df-dju 9938 df-card 9976 df-acn 9979 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ocomp 17318 df-ds 17319 df-0g 17487 df-mre 17630 df-mrc 17631 df-mri 17632 df-acs 17633 df-proset 18351 df-drs 18352 df-poset 18370 df-ipo 18585 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lbs 21091 df-lvec 21119 df-tng 24612 df-dim 33626 |
This theorem is referenced by: rrxdim 33641 |
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