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| Mirrors > Home > MPE Home > Th. List > cramer | Structured version Visualization version GIF version | ||
| Description: Cramer's rule. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the [unique] solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations." If it is assumed that a (unique) solution exists, it can be obtained by Cramer's rule (see also cramerimp 22606). On the other hand, if a vector can be constructed by Cramer's rule, it is a solution of the system of linear equations, so at least one solution exists. The uniqueness is ensured by considering only systems of linear equations whose matrix has a unit (of the underlying ring) as determinant, see matunit 22598 or slesolinv 22600. For fields as underlying rings, this requirement is equivalent to the determinant not being 0. Theorem 4.4 in [Lang] p. 513. This is Metamath 100 proof #97. (Contributed by Alexander van der Vekens, 21-Feb-2019.) (Revised by Alexander van der Vekens, 1-Mar-2019.) |
| Ref | Expression |
|---|---|
| cramer.a | β’ π΄ = (π Mat π ) |
| cramer.b | β’ π΅ = (Baseβπ΄) |
| cramer.v | β’ π = ((Baseβπ ) βm π) |
| cramer.d | β’ π· = (π maDet π ) |
| cramer.x | β’ Β· = (π maVecMul β¨π, πβ©) |
| cramer.q | β’ / = (/rβπ ) |
| Ref | Expression |
|---|---|
| cramer | β’ (((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β (π = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) β (π Β· π) = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm3.22 458 | . . 3 β’ ((π β CRing β§ π β β ) β (π β β β§ π β CRing)) | |
| 2 | cramer.a | . . . 4 β’ π΄ = (π Mat π ) | |
| 3 | cramer.b | . . . 4 β’ π΅ = (Baseβπ΄) | |
| 4 | cramer.v | . . . 4 β’ π = ((Baseβπ ) βm π) | |
| 5 | cramer.d | . . . 4 β’ π· = (π maDet π ) | |
| 6 | cramer.x | . . . 4 β’ Β· = (π maVecMul β¨π, πβ©) | |
| 7 | cramer.q | . . . 4 β’ / = (/rβπ ) | |
| 8 | 2, 3, 4, 5, 6, 7 | cramerlem3 22609 | . . 3 β’ (((π β β β§ π β CRing) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β (π = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) β (π Β· π) = π)) |
| 9 | 1, 8 | syl3an1 1160 | . 2 β’ (((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β (π = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) β (π Β· π) = π)) |
| 10 | simpl1l 1221 | . . . 4 β’ ((((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β§ (π Β· π) = π) β π β CRing) | |
| 11 | simpl2 1189 | . . . 4 β’ ((((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β§ (π Β· π) = π) β (π β π΅ β§ π β π)) | |
| 12 | simpl3 1190 | . . . 4 β’ ((((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β§ (π Β· π) = π) β (π·βπ) β (Unitβπ )) | |
| 13 | crngring 20190 | . . . . . . . 8 β’ (π β CRing β π β Ring) | |
| 14 | 13 | anim1ci 614 | . . . . . . 7 β’ ((π β CRing β§ π β β ) β (π β β β§ π β Ring)) |
| 15 | 14 | anim1i 613 | . . . . . 6 β’ (((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π)) β ((π β β β§ π β Ring) β§ (π β π΅ β§ π β π))) |
| 16 | 15 | 3adant3 1129 | . . . . 5 β’ (((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β ((π β β β§ π β Ring) β§ (π β π΅ β§ π β π))) |
| 17 | 2, 3, 4, 6 | slesolvec 22599 | . . . . . 6 β’ (((π β β β§ π β Ring) β§ (π β π΅ β§ π β π)) β ((π Β· π) = π β π β π)) |
| 18 | 17 | imp 405 | . . . . 5 β’ ((((π β β β§ π β Ring) β§ (π β π΅ β§ π β π)) β§ (π Β· π) = π) β π β π) |
| 19 | 16, 18 | sylan 578 | . . . 4 β’ ((((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β§ (π Β· π) = π) β π β π) |
| 20 | simpr 483 | . . . 4 β’ ((((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β§ (π Β· π) = π) β (π Β· π) = π) | |
| 21 | 2, 3, 4, 5, 6, 7 | cramerlem1 22607 | . . . 4 β’ ((π β CRing β§ (π β π΅ β§ π β π) β§ ((π·βπ) β (Unitβπ ) β§ π β π β§ (π Β· π) = π)) β π = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ)))) |
| 22 | 10, 11, 12, 19, 20, 21 | syl113anc 1379 | . . 3 β’ ((((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β§ (π Β· π) = π) β π = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ)))) |
| 23 | 22 | ex 411 | . 2 β’ (((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β ((π Β· π) = π β π = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))))) |
| 24 | 9, 23 | impbid 211 | 1 β’ (((π β CRing β§ π β β ) β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β (π = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) β (π Β· π) = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2936 β c0 4324 β¨cop 4636 β¦ cmpt 5233 βcfv 6551 (class class class)co 7424 βm cmap 8849 Basecbs 17185 Ringcrg 20178 CRingccrg 20179 Unitcui 20299 /rcdvr 20344 Mat cmat 22325 maVecMul cmvmul 22460 matRepV cmatrepV 22477 maDet cmdat 22504 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-addf 11223 ax-mulf 11224 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-xnn0 12581 df-z 12595 df-dec 12714 df-uz 12859 df-rp 13013 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-word 14503 df-lsw 14551 df-concat 14559 df-s1 14584 df-substr 14629 df-pfx 14659 df-splice 14738 df-reverse 14747 df-s2 14837 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-0g 17428 df-gsum 17429 df-prds 17434 df-pws 17436 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-submnd 18746 df-efmnd 18826 df-grp 18898 df-minusg 18899 df-sbg 18900 df-mulg 19029 df-subg 19083 df-ghm 19173 df-gim 19218 df-cntz 19273 df-oppg 19302 df-symg 19327 df-pmtr 19402 df-psgn 19451 df-evpm 19452 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-srg 20132 df-ring 20180 df-cring 20181 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-dvr 20345 df-rhm 20416 df-subrng 20488 df-subrg 20513 df-drng 20631 df-lmod 20750 df-lss 20821 df-sra 21063 df-rgmod 21064 df-cnfld 21285 df-zring 21378 df-zrh 21434 df-dsmm 21671 df-frlm 21686 df-assa 21792 df-mamu 22304 df-mat 22326 df-mvmul 22461 df-marrep 22478 df-marepv 22479 df-subma 22497 df-mdet 22505 df-madu 22554 df-minmar1 22555 |
| This theorem is referenced by: (None) |
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