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| Mirrors > Home > MPE Home > Th. List > cramerlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for cramer 22548. (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 614 | . . . 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 | pm3.22 459 | . . . . . . 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 480 | . . . 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 2726 | . . . . 5 β’ (((1rβπ΄)(π matRepV π )π)βπ) = (((1rβπ΄)(π matRepV π )π)βπ) | |
| 12 | eqid 2726 | . . . . 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 22543 | . . . 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 3140 | . 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 8861 | . . . . . 6 β’ (π β ((Baseβπ ) βm π) β π Fn π) | |
| 20 | 19, 10 | eleq2s 2845 | . . . . 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 6890 | . . . 4 β’ (π = π β (π·β((π(π matRepV π )π)βπ)) = (π·β((π(π matRepV π )π)βπ))) | |
| 24 | 23 | oveq1d 7420 | . . 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 7440 | . . 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 7440 | . . 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 7036 | . 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 257 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 Vcvv 3468 β¨cop 4629 β¦ cmpt 5224 Fn wfn 6532 βcfv 6537 (class class class)co 7405 βm cmap 8822 Basecbs 17153 1rcur 20086 CRingccrg 20139 Unitcui 20257 /rcdvr 20302 Mat cmat 22262 maVecMul cmvmul 22397 matRepV cmatrepV 22414 maDet cmdat 22441 |
| 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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192 |
| 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-splice 14706 df-reverse 14715 df-s2 14805 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-efmnd 18794 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-gim 19184 df-cntz 19233 df-oppg 19262 df-symg 19287 df-pmtr 19362 df-psgn 19411 df-evpm 19412 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-dsmm 21627 df-frlm 21642 df-mamu 22241 df-mat 22263 df-mvmul 22398 df-marrep 22415 df-marepv 22416 df-subma 22434 df-mdet 22442 df-minmar1 22492 |
| This theorem is referenced by: cramerlem2 22545 cramer 22548 |
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