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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tnglvec | Structured version Visualization version GIF version |
Description: Augmenting a structure with a norm conserves left vector spaces. (Contributed by Thierry Arnoux, 20-May-2023.) |
Ref | Expression |
---|---|
tnglvec.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
Ref | Expression |
---|---|
tnglvec | ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2732 | . 2 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐺)) | |
2 | tnglvec.t | . . 3 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
3 | eqid 2731 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | tngbas 24384 | . 2 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
5 | eqid 2731 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 2, 5 | tngplusg 24386 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
7 | 6 | oveqdr 7440 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
8 | eqidd 2732 | . 2 ⊢ (𝑁 ∈ 𝑉 → (Scalar‘𝐺) = (Scalar‘𝐺)) | |
9 | eqid 2731 | . . 3 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺) | |
10 | 2, 9 | tngsca 24391 | . 2 ⊢ (𝑁 ∈ 𝑉 → (Scalar‘𝐺) = (Scalar‘𝑇)) |
11 | eqid 2731 | . 2 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺)) | |
12 | eqid 2731 | . . . 4 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺) | |
13 | 2, 12 | tngvsca 24393 | . . 3 ⊢ (𝑁 ∈ 𝑉 → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
14 | 13 | oveqdr 7440 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) = (𝑥( ·𝑠 ‘𝑇)𝑦)) |
15 | 1, 4, 7, 8, 10, 11, 14 | lvecpropd 20929 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Scalarcsca 17207 ·𝑠 cvsca 17208 LVecclvec 20861 toNrmGrp ctng 24320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ds 17226 df-0g 17394 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-mgp 20033 df-ur 20080 df-ring 20133 df-lmod 20620 df-lvec 20862 df-tng 24326 |
This theorem is referenced by: tngdim 33001 |
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