Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 22539). 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 22531 or slesolinv 22533. 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 | pm3.22 459 | . . 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 22542 | . . 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 20148 | . . . . . . . 8 β’ (π β CRing β π β Ring) | |
| 14 | 13 | anim1ci 615 | . . . . . . 7 β’ ((π β CRing β§ π β β ) β (π β β β§ π β Ring)) |
| 15 | 14 | anim1i 614 | . . . . . 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 22532 | . . . . . 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 406 | . . . . 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 579 | . . . 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 484 | . . . 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 22540 | . . . 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 412 | . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 β c0 4317 β¨cop 4629 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 βm cmap 8819 Basecbs 17151 Ringcrg 20136 CRingccrg 20137 Unitcui 20255 /rcdvr 20300 Mat cmat 22258 maVecMul cmvmul 22393 matRepV cmatrepV 22410 maDet cmdat 22437 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14031 df-hash 14294 df-word 14469 df-lsw 14517 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14704 df-reverse 14713 df-s2 14803 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-efmnd 18792 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19048 df-ghm 19137 df-gim 19182 df-cntz 19231 df-oppg 19260 df-symg 19285 df-pmtr 19360 df-psgn 19409 df-evpm 19410 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-srg 20090 df-ring 20138 df-cring 20139 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-rhm 20372 df-subrng 20444 df-subrg 20469 df-drng 20587 df-lmod 20706 df-lss 20777 df-sra 21019 df-rgmod 21020 df-cnfld 21237 df-zring 21330 df-zrh 21386 df-dsmm 21623 df-frlm 21638 df-assa 21744 df-mamu 22237 df-mat 22259 df-mvmul 22394 df-marrep 22411 df-marepv 22412 df-subma 22430 df-mdet 22438 df-madu 22487 df-minmar1 22488 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |