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| Mirrors > Home > MPE Home > Th. List > cramerlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for cramer 22192. (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 1136 | . . . . 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 615 | . . . 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 1192 | . . . 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 | pm3.22 460 | . . . . . . 7 β’ (((π·βπ) β (Unitβπ ) β§ (π Β· π) = π) β ((π Β· π) = π β§ (π·βπ) β (Unitβπ ))) | |
| 5 | 4 | 3adant2 1131 | . . . . . 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 1135 | . . . . 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 481 | . . . 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 2732 | . . . . 5 β’ (((1rβπ΄)(π matRepV π )π)βπ) = (((1rβπ΄)(π matRepV π )π)βπ) | |
| 12 | eqid 2732 | . . . . 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 22187 | . . . 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 1371 | . . 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 3146 | . 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 8858 | . . . . . 6 β’ (π β ((Baseβπ ) βm π) β π Fn π) | |
| 20 | 19, 10 | eleq2s 2851 | . . . . 5 β’ (π β π β π Fn π) |
| 21 | 20 | 3ad2ant2 1134 | . . . 4 β’ (((π·βπ) β (Unitβπ ) β§ π β π β§ (π Β· π) = π) β π Fn π) |
| 22 | 21 | 3ad2ant3 1135 | . . 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 6896 | . . . 4 β’ (π = π β (π·β((π(π matRepV π )π)βπ)) = (π·β((π(π matRepV π )π)βπ))) | |
| 24 | 23 | oveq1d 7423 | . . 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 7443 | . . 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 7443 | . . 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 7042 | . 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 β¨cop 4634 β¦ cmpt 5231 Fn wfn 6538 βcfv 6543 (class class class)co 7408 βm cmap 8819 Basecbs 17143 1rcur 20003 CRingccrg 20056 Unitcui 20168 /rcdvr 20213 Mat cmat 21906 maVecMul cmvmul 22041 matRepV cmatrepV 22058 maDet cmdat 22085 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 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 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-xnn0 12544 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-word 14464 df-lsw 14512 df-concat 14520 df-s1 14545 df-substr 14590 df-pfx 14620 df-splice 14699 df-reverse 14708 df-s2 14798 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-efmnd 18749 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-gim 19132 df-cntz 19180 df-oppg 19209 df-symg 19234 df-pmtr 19309 df-psgn 19358 df-evpm 19359 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-srg 20009 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-rnghom 20250 df-subrg 20316 df-drng 20358 df-lmod 20472 df-lss 20542 df-sra 20784 df-rgmod 20785 df-cnfld 20944 df-zring 21017 df-zrh 21052 df-dsmm 21286 df-frlm 21301 df-mamu 21885 df-mat 21907 df-mvmul 22042 df-marrep 22059 df-marepv 22060 df-subma 22078 df-mdet 22086 df-minmar1 22136 |
| This theorem is referenced by: cramerlem2 22189 cramer 22192 |
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