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Mirrors > Home > MPE Home > Th. List > Mathboxes > frlmdim | Structured version Visualization version GIF version |
Description: Dimension of a free left module. (Contributed by Thierry Arnoux, 20-May-2023.) |
Ref | Expression |
---|---|
frlmdim.f | β’ πΉ = (π freeLMod πΌ) |
Ref | Expression |
---|---|
frlmdim | β’ ((π β DivRing β§ πΌ β π) β (dimβπΉ) = (β―βπΌ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmdim.f | . . . 4 β’ πΉ = (π freeLMod πΌ) | |
2 | 1 | frlmlvec 21190 | . . 3 β’ ((π β DivRing β§ πΌ β π) β πΉ β LVec) |
3 | drngring 20226 | . . . 4 β’ (π β DivRing β π β Ring) | |
4 | eqid 2733 | . . . . 5 β’ (π unitVec πΌ) = (π unitVec πΌ) | |
5 | eqid 2733 | . . . . 5 β’ (LBasisβπΉ) = (LBasisβπΉ) | |
6 | 1, 4, 5 | frlmlbs 21226 | . . . 4 β’ ((π β Ring β§ πΌ β π) β ran (π unitVec πΌ) β (LBasisβπΉ)) |
7 | 3, 6 | sylan 581 | . . 3 β’ ((π β DivRing β§ πΌ β π) β ran (π unitVec πΌ) β (LBasisβπΉ)) |
8 | 5 | dimval 32362 | . . 3 β’ ((πΉ β LVec β§ ran (π unitVec πΌ) β (LBasisβπΉ)) β (dimβπΉ) = (β―βran (π unitVec πΌ))) |
9 | 2, 7, 8 | syl2anc 585 | . 2 β’ ((π β DivRing β§ πΌ β π) β (dimβπΉ) = (β―βran (π unitVec πΌ))) |
10 | simpr 486 | . . 3 β’ ((π β DivRing β§ πΌ β π) β πΌ β π) | |
11 | drngnzr 20238 | . . . 4 β’ (π β DivRing β π β NzRing) | |
12 | eqid 2733 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
13 | 4, 1, 12 | uvcf1 21221 | . . . 4 β’ ((π β NzRing β§ πΌ β π) β (π unitVec πΌ):πΌβ1-1β(BaseβπΉ)) |
14 | 11, 13 | sylan 581 | . . 3 β’ ((π β DivRing β§ πΌ β π) β (π unitVec πΌ):πΌβ1-1β(BaseβπΉ)) |
15 | hashf1rn 14261 | . . 3 β’ ((πΌ β π β§ (π unitVec πΌ):πΌβ1-1β(BaseβπΉ)) β (β―β(π unitVec πΌ)) = (β―βran (π unitVec πΌ))) | |
16 | 10, 14, 15 | syl2anc 585 | . 2 β’ ((π β DivRing β§ πΌ β π) β (β―β(π unitVec πΌ)) = (β―βran (π unitVec πΌ))) |
17 | mptexg 7175 | . . . . . . 7 β’ (πΌ β π β (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ ))) β V) | |
18 | 17 | ad2antlr 726 | . . . . . 6 β’ (((π β DivRing β§ πΌ β π) β§ π β πΌ) β (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ ))) β V) |
19 | 18 | ralrimiva 3140 | . . . . 5 β’ ((π β DivRing β§ πΌ β π) β βπ β πΌ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ ))) β V) |
20 | eqid 2733 | . . . . . 6 β’ (π β πΌ β¦ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ )))) = (π β πΌ β¦ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ )))) | |
21 | 20 | fnmpt 6645 | . . . . 5 β’ (βπ β πΌ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ ))) β V β (π β πΌ β¦ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ )))) Fn πΌ) |
22 | 19, 21 | syl 17 | . . . 4 β’ ((π β DivRing β§ πΌ β π) β (π β πΌ β¦ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ )))) Fn πΌ) |
23 | eqid 2733 | . . . . . 6 β’ (1rβπ ) = (1rβπ ) | |
24 | eqid 2733 | . . . . . 6 β’ (0gβπ ) = (0gβπ ) | |
25 | 4, 23, 24 | uvcfval 21213 | . . . . 5 β’ ((π β DivRing β§ πΌ β π) β (π unitVec πΌ) = (π β πΌ β¦ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ ))))) |
26 | 25 | fneq1d 6599 | . . . 4 β’ ((π β DivRing β§ πΌ β π) β ((π unitVec πΌ) Fn πΌ β (π β πΌ β¦ (π β πΌ β¦ if(π = π, (1rβπ ), (0gβπ )))) Fn πΌ)) |
27 | 22, 26 | mpbird 257 | . . 3 β’ ((π β DivRing β§ πΌ β π) β (π unitVec πΌ) Fn πΌ) |
28 | hashfn 14284 | . . 3 β’ ((π unitVec πΌ) Fn πΌ β (β―β(π unitVec πΌ)) = (β―βπΌ)) | |
29 | 27, 28 | syl 17 | . 2 β’ ((π β DivRing β§ πΌ β π) β (β―β(π unitVec πΌ)) = (β―βπΌ)) |
30 | 9, 16, 29 | 3eqtr2d 2779 | 1 β’ ((π β DivRing β§ πΌ β π) β (dimβπΉ) = (β―βπΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3447 ifcif 4490 β¦ cmpt 5192 ran crn 5638 Fn wfn 6495 β1-1βwf1 6497 βcfv 6500 (class class class)co 7361 β―chash 14239 Basecbs 17091 0gc0g 17329 1rcur 19921 Ringcrg 19972 NzRingcnzr 20195 DivRingcdr 20219 LBasisclbs 20579 LVecclvec 20607 freeLMod cfrlm 21175 unitVec cuvc 21211 dimcldim 32360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-reg 9536 ax-inf2 9585 ax-ac2 10407 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-r1 9708 df-rank 9709 df-card 9883 df-acn 9886 df-ac 10060 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ocomp 17162 df-ds 17163 df-hom 17165 df-cco 17166 df-0g 17331 df-gsum 17332 df-prds 17337 df-pws 17339 df-mre 17474 df-mrc 17475 df-mri 17476 df-acs 17477 df-proset 18192 df-drs 18193 df-poset 18210 df-ipo 18425 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-ghm 19014 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-nzr 20196 df-drng 20221 df-subrg 20262 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lmhm 20527 df-lbs 20580 df-lvec 20608 df-sra 20678 df-rgmod 20679 df-dsmm 21161 df-frlm 21176 df-uvc 21212 df-dim 32361 |
This theorem is referenced by: rrxdim 32373 matdim 32374 |
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