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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22det | Structured version Visualization version GIF version |
Description: The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.) |
Ref | Expression |
---|---|
lmat22.m | β’ π = (litMatββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) |
lmat22.a | β’ (π β π΄ β π) |
lmat22.b | β’ (π β π΅ β π) |
lmat22.c | β’ (π β πΆ β π) |
lmat22.d | β’ (π β π· β π) |
lmat22det.t | β’ Β· = (.rβπ ) |
lmat22det.s | β’ β = (-gβπ ) |
lmat22det.v | β’ π = (Baseβπ ) |
lmat22det.j | β’ π½ = ((1...2) maDet π ) |
lmat22det.r | β’ (π β π β Ring) |
Ref | Expression |
---|---|
lmat22det | β’ (π β (π½βπ) = ((π΄ Β· π·) β (πΆ Β· π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22det.r | . . 3 β’ (π β π β Ring) | |
2 | lmat22.m | . . . 4 β’ π = (litMatββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) | |
3 | 2nn 12315 | . . . . 5 β’ 2 β β | |
4 | 3 | a1i 11 | . . . 4 β’ (π β 2 β β) |
5 | lmat22.a | . . . . . 6 β’ (π β π΄ β π) | |
6 | lmat22.b | . . . . . 6 β’ (π β π΅ β π) | |
7 | 5, 6 | s2cld 14854 | . . . . 5 β’ (π β β¨βπ΄π΅ββ© β Word π) |
8 | lmat22.c | . . . . . 6 β’ (π β πΆ β π) | |
9 | lmat22.d | . . . . . 6 β’ (π β π· β π) | |
10 | 8, 9 | s2cld 14854 | . . . . 5 β’ (π β β¨βπΆπ·ββ© β Word π) |
11 | 7, 10 | s2cld 14854 | . . . 4 β’ (π β β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ© β Word Word π) |
12 | s2len 14872 | . . . . 5 β’ (β―ββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) = 2 | |
13 | 12 | a1i 11 | . . . 4 β’ (π β (β―ββ¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©) = 2) |
14 | 2, 5, 6, 8, 9 | lmat22lem 33475 | . . . 4 β’ ((π β§ π β (0..^2)) β (β―β(β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ©βπ)) = 2) |
15 | lmat22det.v | . . . 4 β’ π = (Baseβπ ) | |
16 | eqid 2725 | . . . 4 β’ ((1...2) Mat π ) = ((1...2) Mat π ) | |
17 | eqid 2725 | . . . 4 β’ (Baseβ((1...2) Mat π )) = (Baseβ((1...2) Mat π )) | |
18 | 2, 4, 11, 13, 14, 15, 16, 17, 1 | lmatcl 33474 | . . 3 β’ (π β π β (Baseβ((1...2) Mat π ))) |
19 | 2z 12624 | . . . . . 6 β’ 2 β β€ | |
20 | fzval3 13733 | . . . . . 6 β’ (2 β β€ β (1...2) = (1..^(2 + 1))) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 β’ (1...2) = (1..^(2 + 1)) |
22 | 2p1e3 12384 | . . . . . 6 β’ (2 + 1) = 3 | |
23 | 22 | oveq2i 7427 | . . . . 5 β’ (1..^(2 + 1)) = (1..^3) |
24 | fzo13pr 13748 | . . . . 5 β’ (1..^3) = {1, 2} | |
25 | 21, 23, 24 | 3eqtri 2757 | . . . 4 β’ (1...2) = {1, 2} |
26 | lmat22det.j | . . . 4 β’ π½ = ((1...2) maDet π ) | |
27 | lmat22det.s | . . . 4 β’ β = (-gβπ ) | |
28 | lmat22det.t | . . . 4 β’ Β· = (.rβπ ) | |
29 | 25, 26, 16, 17, 27, 28 | m2detleib 22551 | . . 3 β’ ((π β Ring β§ π β (Baseβ((1...2) Mat π ))) β (π½βπ) = (((1π1) Β· (2π2)) β ((2π1) Β· (1π2)))) |
30 | 1, 18, 29 | syl2anc 582 | . 2 β’ (π β (π½βπ) = (((1π1) Β· (2π2)) β ((2π1) Β· (1π2)))) |
31 | 2, 5, 6, 8, 9 | lmat22e11 33476 | . . . 4 β’ (π β (1π1) = π΄) |
32 | 2, 5, 6, 8, 9 | lmat22e22 33479 | . . . 4 β’ (π β (2π2) = π·) |
33 | 31, 32 | oveq12d 7434 | . . 3 β’ (π β ((1π1) Β· (2π2)) = (π΄ Β· π·)) |
34 | 2, 5, 6, 8, 9 | lmat22e21 33478 | . . . 4 β’ (π β (2π1) = πΆ) |
35 | 2, 5, 6, 8, 9 | lmat22e12 33477 | . . . 4 β’ (π β (1π2) = π΅) |
36 | 34, 35 | oveq12d 7434 | . . 3 β’ (π β ((2π1) Β· (1π2)) = (πΆ Β· π΅)) |
37 | 33, 36 | oveq12d 7434 | . 2 β’ (π β (((1π1) Β· (2π2)) β ((2π1) Β· (1π2))) = ((π΄ Β· π·) β (πΆ Β· π΅))) |
38 | 30, 37 | eqtrd 2765 | 1 β’ (π β (π½βπ) = ((π΄ Β· π·) β (πΆ Β· π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {cpr 4626 βcfv 6543 (class class class)co 7416 1c1 11139 + caddc 11141 βcn 12242 2c2 12297 3c3 12298 β€cz 12588 ...cfz 13516 ..^cfzo 13659 β―chash 14321 Word cword 14496 β¨βcs2 14824 Basecbs 17179 .rcmulr 17233 -gcsg 18896 Ringcrg 20177 Mat cmat 22325 maDet cmdat 22504 litMatclmat 33469 |
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-oadd 8489 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-dju 9924 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-fac 14265 df-bc 14294 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-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-sra 21062 df-rgmod 21063 df-cnfld 21284 df-zring 21377 df-zrh 21433 df-dsmm 21670 df-frlm 21685 df-mat 22326 df-mdet 22505 df-lmat 33470 |
This theorem is referenced by: (None) |
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