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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 2772 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐺)) | |
2 | tnglvec.t | . . . 4 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
3 | eqid 2771 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | tngbas 22968 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
5 | 4 | adantl 474 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝑇)) |
6 | ssidd 3873 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) ⊆ (Base‘𝐺)) | |
7 | eqid 2771 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | 2, 7 | tngplusg 22969 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
9 | 8 | adantl 474 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝑇)) |
10 | 9 | oveqdr 7002 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
11 | lveclmod 19612 | . . . 4 ⊢ (𝐺 ∈ LVec → 𝐺 ∈ LMod) | |
12 | eqid 2771 | . . . . . 6 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺) | |
13 | eqid 2771 | . . . . . 6 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺) | |
14 | eqid 2771 | . . . . . 6 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺)) | |
15 | 3, 12, 13, 14 | lmodvscl 19385 | . . . . 5 ⊢ ((𝐺 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
16 | 15 | 3expb 1101 | . . . 4 ⊢ ((𝐺 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
17 | 11, 16 | sylan 572 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
18 | 17 | adantlr 703 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
19 | 2, 13 | tngvsca 22973 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
20 | 19 | adantl 474 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
21 | 20 | oveqdr 7002 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) = (𝑥( ·𝑠 ‘𝑇)𝑦)) |
22 | eqid 2771 | . 2 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
23 | eqidd 2772 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺))) | |
24 | 2, 12 | tngsca 22972 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Scalar‘𝐺) = (Scalar‘𝑇)) |
25 | 24 | adantl 474 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝑇)) |
26 | 25 | fveq2d 6500 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑇))) |
27 | 25 | fveq2d 6500 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘(Scalar‘𝐺)) = (+g‘(Scalar‘𝑇))) |
28 | 27 | oveqdr 7002 | . 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 475 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ LVec) | |
30 | 2 | tnglvec 30671 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec)) |
31 | 30 | biimpac 471 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝑇 ∈ LVec) |
32 | 1, 5, 6, 10, 18, 21, 12, 22, 23, 26, 28, 29, 31 | dimpropd 30668 | 1 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 +gcplusg 16419 Scalarcsca 16422 ·𝑠 cvsca 16423 LModclmod 19368 LVecclvec 19608 toNrmGrp ctng 22906 dimcldim 30660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-reg 8849 ax-inf2 8896 ax-ac2 9681 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-rpss 7265 df-om 7395 df-1st 7499 df-2nd 7500 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-oi 8767 df-r1 8985 df-rank 8986 df-dju 9122 df-card 9160 df-acn 9163 df-ac 9334 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-xnn0 11778 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-hash 13504 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-tset 16438 df-ple 16439 df-ocomp 16440 df-ds 16441 df-0g 16569 df-mre 16727 df-mrc 16728 df-mri 16729 df-acs 16730 df-proset 17408 df-drs 17409 df-poset 17426 df-ipo 17632 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-drng 19239 df-lmod 19370 df-lss 19438 df-lsp 19478 df-lbs 19581 df-lvec 19609 df-tng 22912 df-dim 30661 |
This theorem is referenced by: rrxdim 30673 |
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