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 2821 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐺)) | |
2 | tnglvec.t | . . . 4 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
3 | eqid 2820 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | tngbas 23245 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
5 | 4 | adantl 484 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝑇)) |
6 | ssidd 3983 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘𝐺) ⊆ (Base‘𝐺)) | |
7 | eqid 2820 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | 2, 7 | tngplusg 23246 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝑇)) |
10 | 9 | oveqdr 7177 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
11 | lveclmod 19873 | . . . 4 ⊢ (𝐺 ∈ LVec → 𝐺 ∈ LMod) | |
12 | eqid 2820 | . . . . . 6 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺) | |
13 | eqid 2820 | . . . . . 6 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺) | |
14 | eqid 2820 | . . . . . 6 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺)) | |
15 | 3, 12, 13, 14 | lmodvscl 19646 | . . . . 5 ⊢ ((𝐺 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
16 | 15 | 3expb 1115 | . . . 4 ⊢ ((𝐺 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) ∈ (Base‘𝐺)) |
17 | 11, 16 | sylan 582 | . . 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 23250 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
20 | 19 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝑇)) |
21 | 20 | oveqdr 7177 | . 2 ⊢ (((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥( ·𝑠 ‘𝐺)𝑦) = (𝑥( ·𝑠 ‘𝑇)𝑦)) |
22 | eqid 2820 | . 2 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
23 | eqidd 2821 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺))) | |
24 | 2, 12 | tngsca 23249 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Scalar‘𝐺) = (Scalar‘𝑇)) |
25 | 24 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝑇)) |
26 | 25 | fveq2d 6667 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑇))) |
27 | 25 | fveq2d 6667 | . . 3 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (+g‘(Scalar‘𝐺)) = (+g‘(Scalar‘𝑇))) |
28 | 27 | oveqdr 7177 | . 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 485 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ LVec) | |
30 | 2 | tnglvec 31034 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec)) |
31 | 30 | biimpac 481 | . 2 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → 𝑇 ∈ LVec) |
32 | 1, 5, 6, 10, 18, 21, 12, 22, 23, 26, 28, 29, 31 | dimpropd 31031 | 1 ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 Scalarcsca 16563 ·𝑠 cvsca 16564 LModclmod 19629 LVecclvec 19869 toNrmGrp ctng 23183 dimcldim 31023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-reg 9049 ax-inf2 9097 ax-ac2 9878 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-rpss 7442 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-oi 8967 df-r1 9186 df-rank 9187 df-dju 9323 df-card 9361 df-acn 9364 df-ac 9535 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-hash 13688 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-tset 16579 df-ple 16580 df-ocomp 16581 df-ds 16582 df-0g 16710 df-mre 16852 df-mrc 16853 df-mri 16854 df-acs 16855 df-proset 17533 df-drs 17534 df-poset 17551 df-ipo 17757 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-drng 19499 df-lmod 19631 df-lss 19699 df-lsp 19739 df-lbs 19842 df-lvec 19870 df-tng 23189 df-dim 31024 |
This theorem is referenced by: rrxdim 31036 |
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