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| Mirrors > Home > MPE Home > Th. List > cramerlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for cramer 22609. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 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 |
|---|---|
| cramerlem2 | β’ ((π β CRing β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β βπ§ β π ((π Β· π§) = π β π§ = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpll1 1209 | . . . 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 β CRing) | |
| 2 | simpll2 1210 | . . . 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 β π΅ β§ π β π)) | |
| 3 | simpll3 1211 | . . . 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·βπ) β (Unitβπ )) | |
| 4 | simplr 767 | . . . 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§ β π) | |
| 5 | simpr 483 | . . . 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 Β· π§) = π) | |
| 6 | cramer.a | . . . . 5 β’ π΄ = (π Mat π ) | |
| 7 | cramer.b | . . . . 5 β’ π΅ = (Baseβπ΄) | |
| 8 | cramer.v | . . . . 5 β’ π = ((Baseβπ ) βm π) | |
| 9 | cramer.d | . . . . 5 β’ π· = (π maDet π ) | |
| 10 | cramer.x | . . . . 5 β’ Β· = (π maVecMul β¨π, πβ©) | |
| 11 | cramer.q | . . . . 5 β’ / = (/rβπ ) | |
| 12 | 6, 7, 8, 9, 10, 11 | cramerlem1 22605 | . . . 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)))) |
| 13 | 1, 2, 3, 4, 5, 12 | syl113anc 1379 | . . 3 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)))) |
| 14 | 13 | ex 411 | . 2 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))))) |
| 15 | 14 | ralrimiva 3136 | 1 β’ ((π β CRing β§ (π β π΅ β§ π β π) β§ (π·βπ) β (Unitβπ )) β βπ§ β π ((π Β· π§) = π β π§ = (π β π β¦ ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 β¨cop 4630 β¦ cmpt 5226 βcfv 6542 (class class class)co 7415 βm cmap 8841 Basecbs 17177 CRingccrg 20176 Unitcui 20296 /rcdvr 20341 Mat cmat 22323 maVecMul cmvmul 22458 matRepV cmatrepV 22475 maDet cmdat 22502 |
| 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-addf 11215 ax-mulf 11216 |
| 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-xnn0 12573 df-z 12587 df-dec 12706 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-word 14495 df-lsw 14543 df-concat 14551 df-s1 14576 df-substr 14621 df-pfx 14651 df-splice 14730 df-reverse 14739 df-s2 14829 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-efmnd 18823 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-gim 19215 df-cntz 19270 df-oppg 19299 df-symg 19324 df-pmtr 19399 df-psgn 19448 df-evpm 19449 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-srg 20129 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-drng 20628 df-lmod 20747 df-lss 20818 df-sra 21060 df-rgmod 21061 df-cnfld 21282 df-zring 21375 df-zrh 21431 df-dsmm 21668 df-frlm 21683 df-mamu 22307 df-mat 22324 df-mvmul 22459 df-marrep 22476 df-marepv 22477 df-subma 22495 df-mdet 22503 df-minmar1 22553 |
| This theorem is referenced by: cramerlem3 22607 |
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