Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > cramerlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for cramer 22611. (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 |
|---|---|
| cramerlem1 | 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)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1133 | . . . . 5 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 | 1 | anim1i 613 | . . . 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 CRing β§ π β π)) |
| 3 | simpl2 1189 | . . . 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΅ β§ π β π)) | |
| 4 | pm3.22 458 | . . . . . . 7 β’ (((π·βπ) β (Unitβπ ) β§ (π Β· π) = π) β ((π Β· π) = π β§ (π·βπ) β (Unitβπ ))) | |
| 5 | 4 | 3adant2 1128 | . . . . . 6 β’ (((π·βπ) β (Unitβπ ) β§ π β π β§ (π Β· π) = π) β ((π Β· π) = π β§ (π·βπ) β (Unitβπ ))) |
| 6 | 5 | 3ad2ant3 1132 | . . . . 5 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 β§ (π·βπ) β (Unitβπ ))) |
| 7 | 6 | adantr 479 | . . . 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) = π β§ (π·βπ) β (Unitβπ ))) |
| 8 | cramer.a | . . . . 5 β’ π΄ = (π Mat π ) | |
| 9 | cramer.b | . . . . 5 β’ π΅ = (Baseβπ΄) | |
| 10 | cramer.v | . . . . 5 β’ π = ((Baseβπ ) βm π) | |
| 11 | eqid 2725 | . . . . 5 β’ (((1rβπ΄)(π matRepV π )π)βπ) = (((1rβπ΄)(π matRepV π )π)βπ) | |
| 12 | eqid 2725 | . . . . 5 β’ ((π(π matRepV π )π)βπ) = ((π(π matRepV π )π)βπ) | |
| 13 | cramer.x | . . . . 5 β’ Β· = (π maVecMul β¨π, πβ©) | |
| 14 | cramer.d | . . . . 5 β’ π· = (π maDet π ) | |
| 15 | cramer.q | . . . . 5 β’ / = (/rβπ ) | |
| 16 | 8, 9, 10, 11, 12, 13, 14, 15 | cramerimp 22606 | . . . 4 β’ (((π β CRing β§ π β π) β§ (π β π΅ β§ π β π) β§ ((π Β· π) = π β§ (π·βπ) β (Unitβπ ))) β (πβπ) = ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) |
| 17 | 2, 3, 7, 16 | syl3anc 1368 | . . 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 β π) β (πβπ) = ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) |
| 18 | 17 | ralrimiva 3136 | . 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 βπ β π (πβπ) = ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) |
| 19 | elmapfn 8882 | . . . . . 6 β’ (π β ((Baseβπ ) βm π) β π Fn π) | |
| 20 | 19, 10 | eleq2s 2843 | . . . . 5 β’ (π β π β π Fn π) |
| 21 | 20 | 3ad2ant2 1131 | . . . 4 β’ (((π·βπ) β (Unitβπ ) β§ π β π β§ (π Β· π) = π) β π Fn π) |
| 22 | 21 | 3ad2ant3 1132 | . . 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 Fn π) |
| 23 | 2fveq3 6897 | . . . 4 β’ (π = π β (π·β((π(π matRepV π )π)βπ)) = (π·β((π(π matRepV π )π)βπ))) | |
| 24 | 23 | oveq1d 7431 | . . 3 β’ (π = π β ((π·β((π(π matRepV π )π)βπ)) / (π·βπ)) = ((π·β((π(π matRepV π )π)βπ)) / (π·βπ))) |
| 25 | ovexd 7451 | . . 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 β π) β ((π·β((π(π matRepV π )π)βπ)) / (π·βπ)) β V) | |
| 26 | ovexd 7451 | . . 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 β π) β ((π·β((π(π matRepV π )π)βπ)) / (π·βπ)) β V) | |
| 27 | 22, 24, 25, 26 | fnmptfvd 7045 | . 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))) β βπ β π (πβπ) = ((π·β((π(π matRepV π )π)βπ)) / (π·βπ)))) |
| 28 | 18, 27 | mpbird 256 | 1 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)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 Vcvv 3463 β¨cop 4630 β¦ cmpt 5226 Fn wfn 6538 βcfv 6543 (class class class)co 7416 βm cmap 8843 Basecbs 17179 1rcur 20125 CRingccrg 20178 Unitcui 20298 /rcdvr 20343 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218 |
| 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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-splice 14732 df-reverse 14741 df-s2 14831 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-efmnd 18825 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-gim 19217 df-cntz 19272 df-oppg 19301 df-symg 19326 df-pmtr 19401 df-psgn 19450 df-evpm 19451 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-srg 20131 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-drng 20630 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-cnfld 21284 df-zring 21377 df-zrh 21433 df-dsmm 21670 df-frlm 21685 df-mamu 22309 df-mat 22326 df-mvmul 22461 df-marrep 22478 df-marepv 22479 df-subma 22497 df-mdet 22505 df-minmar1 22555 |
| This theorem is referenced by: cramerlem2 22608 cramer 22611 |
| Copyright terms: Public domain | W3C validator |