|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lmimdim | Structured version Visualization version GIF version | ||
| Description: Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| lmimdim.1 | ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMIso 𝑇)) | 
| lmimdim.2 | ⊢ (𝜑 → 𝑆 ∈ LVec) | 
| Ref | Expression | 
|---|---|
| lmimdim | ⊢ (𝜑 → (dim‘𝑆) = (dim‘𝑇)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lmimdim.2 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ LVec) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (LBasis‘𝑆) = (LBasis‘𝑆) | |
| 3 | 2 | lbsex 21168 | . . . 4 ⊢ (𝑆 ∈ LVec → (LBasis‘𝑆) ≠ ∅) | 
| 4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → (LBasis‘𝑆) ≠ ∅) | 
| 5 | n0 4352 | . . 3 ⊢ ((LBasis‘𝑆) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (LBasis‘𝑆)) | |
| 6 | 4, 5 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑏 𝑏 ∈ (LBasis‘𝑆)) | 
| 7 | lmimdim.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMIso 𝑇)) | |
| 8 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → 𝐹 ∈ (𝑆 LMIso 𝑇)) | 
| 9 | 8 | resexd 6045 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (𝐹 ↾ 𝑏) ∈ V) | 
| 10 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 11 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
| 12 | 10, 11 | lmimf1o 21063 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) | 
| 13 | f1of1 6846 | . . . . . . 7 ⊢ (𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇) → 𝐹:(Base‘𝑆)–1-1→(Base‘𝑇)) | |
| 14 | 8, 12, 13 | 3syl 18 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → 𝐹:(Base‘𝑆)–1-1→(Base‘𝑇)) | 
| 15 | 10, 2 | lbsss 21077 | . . . . . . 7 ⊢ (𝑏 ∈ (LBasis‘𝑆) → 𝑏 ⊆ (Base‘𝑆)) | 
| 16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → 𝑏 ⊆ (Base‘𝑆)) | 
| 17 | f1ssres 6810 | . . . . . 6 ⊢ ((𝐹:(Base‘𝑆)–1-1→(Base‘𝑇) ∧ 𝑏 ⊆ (Base‘𝑆)) → (𝐹 ↾ 𝑏):𝑏–1-1→(Base‘𝑇)) | |
| 18 | 14, 16, 17 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (𝐹 ↾ 𝑏):𝑏–1-1→(Base‘𝑇)) | 
| 19 | hashf1dmrn 14483 | . . . . 5 ⊢ (((𝐹 ↾ 𝑏) ∈ V ∧ (𝐹 ↾ 𝑏):𝑏–1-1→(Base‘𝑇)) → (♯‘𝑏) = (♯‘ran (𝐹 ↾ 𝑏))) | |
| 20 | 9, 18, 19 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (♯‘𝑏) = (♯‘ran (𝐹 ↾ 𝑏))) | 
| 21 | df-ima 5697 | . . . . 5 ⊢ (𝐹 “ 𝑏) = ran (𝐹 ↾ 𝑏) | |
| 22 | 21 | fveq2i 6908 | . . . 4 ⊢ (♯‘(𝐹 “ 𝑏)) = (♯‘ran (𝐹 ↾ 𝑏)) | 
| 23 | 20, 22 | eqtr4di 2794 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (♯‘𝑏) = (♯‘(𝐹 “ 𝑏))) | 
| 24 | 2 | dimval 33652 | . . . 4 ⊢ ((𝑆 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑆)) → (dim‘𝑆) = (♯‘𝑏)) | 
| 25 | 1, 24 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (dim‘𝑆) = (♯‘𝑏)) | 
| 26 | lmimlmhm 21064 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | |
| 27 | 7, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) | 
| 28 | lmhmlvec 21110 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec)) | |
| 29 | 28 | biimpa 476 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑆 ∈ LVec) → 𝑇 ∈ LVec) | 
| 30 | 27, 1, 29 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ LVec) | 
| 31 | 30 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → 𝑇 ∈ LVec) | 
| 32 | eqid 2736 | . . . . . 6 ⊢ (LBasis‘𝑇) = (LBasis‘𝑇) | |
| 33 | 2, 32 | lmimlbs 21857 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝑏 ∈ (LBasis‘𝑆)) → (𝐹 “ 𝑏) ∈ (LBasis‘𝑇)) | 
| 34 | 7, 33 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (𝐹 “ 𝑏) ∈ (LBasis‘𝑇)) | 
| 35 | 32 | dimval 33652 | . . . 4 ⊢ ((𝑇 ∈ LVec ∧ (𝐹 “ 𝑏) ∈ (LBasis‘𝑇)) → (dim‘𝑇) = (♯‘(𝐹 “ 𝑏))) | 
| 36 | 31, 34, 35 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (dim‘𝑇) = (♯‘(𝐹 “ 𝑏))) | 
| 37 | 23, 25, 36 | 3eqtr4d 2786 | . 2 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘𝑆)) → (dim‘𝑆) = (dim‘𝑇)) | 
| 38 | 6, 37 | exlimddv 1934 | 1 ⊢ (𝜑 → (dim‘𝑆) = (dim‘𝑇)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 ran crn 5685 ↾ cres 5686 “ cima 5687 –1-1→wf1 6557 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 ♯chash 14370 Basecbs 17248 LMHom clmhm 21019 LMIso clmim 21020 LBasisclbs 21074 LVecclvec 21102 dimcldim 33650 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-reg 9633 ax-inf2 9682 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-rpss 7744 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-oi 9551 df-r1 9805 df-rank 9806 df-dju 9942 df-card 9980 df-acn 9983 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-tset 17317 df-ple 17318 df-ocomp 17319 df-0g 17487 df-mre 17630 df-mrc 17631 df-mri 17632 df-acs 17633 df-proset 18341 df-drs 18342 df-poset 18360 df-ipo 18574 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-ghm 19232 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lmhm 21022 df-lmim 21023 df-lbs 21075 df-lvec 21103 df-lindf 21827 df-linds 21828 df-dim 33651 | 
| This theorem is referenced by: lmicdim 33656 algextdeglem4 33762 | 
| Copyright terms: Public domain | W3C validator |