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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 2726 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐺)) | |
2 | tnglvec.t | . . . 4 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
3 | eqid 2725 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | tngbas 24595 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
5 | 4 | adantl 480 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝑇)) |
6 | ssidd 4000 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) ⊆ (Base‘𝐺)) | |
7 | eqid 2725 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | 2, 7 | tngplusg 24597 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
9 | 8 | adantl 480 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝑇)) |
10 | 9 | oveqdr 7447 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
11 | lveclmod 21003 | . . . 4 ⊢ (𝐺 ∈ LVec → 𝐺 ∈ LMod) | |
12 | eqid 2725 | . . . . . 6 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺) | |
13 | eqid 2725 | . . . . . 6 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺) | |
14 | eqid 2725 | . . . . . 6 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺)) | |
15 | 3, 12, 13, 14 | lmodvscl 20773 | . . . . 5 ⊢ ((𝐺 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
16 | 15 | 3expb 1117 | . . . 4 ⊢ ((𝐺 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
17 | 11, 16 | sylan 578 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
18 | 17 | adantlr 713 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
19 | 2, 13 | tngvsca 24604 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
20 | 19 | adantl 480 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
21 | 20 | oveqdr 7447 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) = (𝑥( ·𝑠 ‘𝑇)𝑦)) |
22 | eqid 2725 | . 2 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
23 | eqidd 2726 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺))) | |
24 | 2, 12 | tngsca 24602 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Scalar‘𝐺) = (Scalar‘𝑇)) |
25 | 24 | adantl 480 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝑇)) |
26 | 25 | fveq2d 6900 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑇))) |
27 | 25 | fveq2d 6900 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘(Scalar‘𝐺)) = (+g‘(Scalar‘𝑇))) |
28 | 27 | oveqdr 7447 | . 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 481 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ LVec) | |
30 | 2 | tnglvec 33441 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec)) |
31 | 30 | biimpac 477 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝑇 ∈ LVec) |
32 | 1, 5, 6, 10, 18, 21, 12, 22, 23, 26, 28, 29, 31 | dimpropd 33437 | 1 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 +gcplusg 17236 Scalarcsca 17239 ·𝑠 cvsca 17240 LModclmod 20755 LVecclvec 20999 toNrmGrp ctng 24531 dimcldim 33427 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-reg 9617 ax-inf2 9666 ax-ac2 10488 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-rpss 7729 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9535 df-r1 9789 df-rank 9790 df-dju 9926 df-card 9964 df-acn 9967 df-ac 10141 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-xnn0 12578 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ocomp 17257 df-ds 17258 df-0g 17426 df-mre 17569 df-mrc 17570 df-mri 17571 df-acs 17572 df-proset 18290 df-drs 18291 df-poset 18308 df-ipo 18523 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lbs 20972 df-lvec 21000 df-tng 24537 df-dim 33428 |
This theorem is referenced by: rrxdim 33443 |
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