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Mirrors > Home > MPE Home > Th. List > lvecdim | Structured version Visualization version GIF version |
Description: The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 20903 and lbsacsbs 20915 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 18517. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
lvecdim.1 | β’ π½ = (LBasisβπ) |
Ref | Expression |
---|---|
lvecdim | β’ ((π β LVec β§ π β π½ β§ π β π½) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
2 | eqid 2731 | . . . . 5 β’ (mrClsβ(LSubSpβπ)) = (mrClsβ(LSubSpβπ)) | |
3 | eqid 2731 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
4 | 1, 2, 3 | lssacsex 20903 | . . . 4 β’ (π β LVec β ((LSubSpβπ) β (ACSβ(Baseβπ)) β§ βπ₯ β π« (Baseβπ)βπ¦ β (Baseβπ)βπ§ β (((mrClsβ(LSubSpβπ))β(π₯ βͺ {π¦})) β ((mrClsβ(LSubSpβπ))βπ₯))π¦ β ((mrClsβ(LSubSpβπ))β(π₯ βͺ {π§})))) |
5 | 4 | 3ad2ant1 1132 | . . 3 β’ ((π β LVec β§ π β π½ β§ π β π½) β ((LSubSpβπ) β (ACSβ(Baseβπ)) β§ βπ₯ β π« (Baseβπ)βπ¦ β (Baseβπ)βπ§ β (((mrClsβ(LSubSpβπ))β(π₯ βͺ {π¦})) β ((mrClsβ(LSubSpβπ))βπ₯))π¦ β ((mrClsβ(LSubSpβπ))β(π₯ βͺ {π§})))) |
6 | 5 | simpld 494 | . 2 β’ ((π β LVec β§ π β π½ β§ π β π½) β (LSubSpβπ) β (ACSβ(Baseβπ))) |
7 | eqid 2731 | . 2 β’ (mrIndβ(LSubSpβπ)) = (mrIndβ(LSubSpβπ)) | |
8 | 5 | simprd 495 | . 2 β’ ((π β LVec β§ π β π½ β§ π β π½) β βπ₯ β π« (Baseβπ)βπ¦ β (Baseβπ)βπ§ β (((mrClsβ(LSubSpβπ))β(π₯ βͺ {π¦})) β ((mrClsβ(LSubSpβπ))βπ₯))π¦ β ((mrClsβ(LSubSpβπ))β(π₯ βͺ {π§}))) |
9 | simp2 1136 | . . . 4 β’ ((π β LVec β§ π β π½ β§ π β π½) β π β π½) | |
10 | lvecdim.1 | . . . . . 6 β’ π½ = (LBasisβπ) | |
11 | 1, 2, 3, 7, 10 | lbsacsbs 20915 | . . . . 5 β’ (π β LVec β (π β π½ β (π β (mrIndβ(LSubSpβπ)) β§ ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ)))) |
12 | 11 | 3ad2ant1 1132 | . . . 4 β’ ((π β LVec β§ π β π½ β§ π β π½) β (π β π½ β (π β (mrIndβ(LSubSpβπ)) β§ ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ)))) |
13 | 9, 12 | mpbid 231 | . . 3 β’ ((π β LVec β§ π β π½ β§ π β π½) β (π β (mrIndβ(LSubSpβπ)) β§ ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ))) |
14 | 13 | simpld 494 | . 2 β’ ((π β LVec β§ π β π½ β§ π β π½) β π β (mrIndβ(LSubSpβπ))) |
15 | simp3 1137 | . . . 4 β’ ((π β LVec β§ π β π½ β§ π β π½) β π β π½) | |
16 | 1, 2, 3, 7, 10 | lbsacsbs 20915 | . . . . 5 β’ (π β LVec β (π β π½ β (π β (mrIndβ(LSubSpβπ)) β§ ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ)))) |
17 | 16 | 3ad2ant1 1132 | . . . 4 β’ ((π β LVec β§ π β π½ β§ π β π½) β (π β π½ β (π β (mrIndβ(LSubSpβπ)) β§ ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ)))) |
18 | 15, 17 | mpbid 231 | . . 3 β’ ((π β LVec β§ π β π½ β§ π β π½) β (π β (mrIndβ(LSubSpβπ)) β§ ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ))) |
19 | 18 | simpld 494 | . 2 β’ ((π β LVec β§ π β π½ β§ π β π½) β π β (mrIndβ(LSubSpβπ))) |
20 | 13 | simprd 495 | . . 3 β’ ((π β LVec β§ π β π½ β§ π β π½) β ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ)) |
21 | 18 | simprd 495 | . . 3 β’ ((π β LVec β§ π β π½ β§ π β π½) β ((mrClsβ(LSubSpβπ))βπ) = (Baseβπ)) |
22 | 20, 21 | eqtr4d 2774 | . 2 β’ ((π β LVec β§ π β π½ β§ π β π½) β ((mrClsβ(LSubSpβπ))βπ) = ((mrClsβ(LSubSpβπ))βπ)) |
23 | 6, 2, 7, 8, 14, 19, 22 | acsexdimd 18517 | 1 β’ ((π β LVec β§ π β π½ β§ π β π½) β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3060 β cdif 3945 βͺ cun 3946 π« cpw 4602 {csn 4628 class class class wbr 5148 βcfv 6543 β cen 8939 Basecbs 17149 mrClscmrc 17532 mrIndcmri 17533 ACScacs 17534 LSubSpclss 20687 LBasisclbs 20830 LVecclvec 20858 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-reg 9590 ax-inf2 9639 ax-ac2 10461 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
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-int 4951 df-iun 4999 df-iin 5000 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-se 5632 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-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-oi 9508 df-r1 9762 df-rank 9763 df-card 9937 df-acn 9940 df-ac 10114 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-tset 17221 df-ple 17222 df-ocomp 17223 df-0g 17392 df-mre 17535 df-mrc 17536 df-mri 17537 df-acs 17538 df-proset 18253 df-drs 18254 df-poset 18271 df-ipo 18486 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lbs 20831 df-lvec 20859 |
This theorem is referenced by: dimval 32974 |
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