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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 12231 | . . . . 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 14766 | . . . . 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 14766 | . . . . 5 β’ (π β β¨βπΆπ·ββ© β Word π) |
11 | 7, 10 | s2cld 14766 | . . . 4 β’ (π β β¨ββ¨βπ΄π΅ββ©β¨βπΆπ·ββ©ββ© β Word Word π) |
12 | s2len 14784 | . . . . 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 32455 | . . . 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 2733 | . . . 4 β’ ((1...2) Mat π ) = ((1...2) Mat π ) | |
17 | eqid 2733 | . . . 4 β’ (Baseβ((1...2) Mat π )) = (Baseβ((1...2) Mat π )) | |
18 | 2, 4, 11, 13, 14, 15, 16, 17, 1 | lmatcl 32454 | . . 3 β’ (π β π β (Baseβ((1...2) Mat π ))) |
19 | 2z 12540 | . . . . . 6 β’ 2 β β€ | |
20 | fzval3 13647 | . . . . . 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 12300 | . . . . . 6 β’ (2 + 1) = 3 | |
23 | 22 | oveq2i 7369 | . . . . 5 β’ (1..^(2 + 1)) = (1..^3) |
24 | fzo13pr 13662 | . . . . 5 β’ (1..^3) = {1, 2} | |
25 | 21, 23, 24 | 3eqtri 2765 | . . . 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 21996 | . . 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 585 | . 2 β’ (π β (π½βπ) = (((1π1) Β· (2π2)) β ((2π1) Β· (1π2)))) |
31 | 2, 5, 6, 8, 9 | lmat22e11 32456 | . . . 4 β’ (π β (1π1) = π΄) |
32 | 2, 5, 6, 8, 9 | lmat22e22 32459 | . . . 4 β’ (π β (2π2) = π·) |
33 | 31, 32 | oveq12d 7376 | . . 3 β’ (π β ((1π1) Β· (2π2)) = (π΄ Β· π·)) |
34 | 2, 5, 6, 8, 9 | lmat22e21 32458 | . . . 4 β’ (π β (2π1) = πΆ) |
35 | 2, 5, 6, 8, 9 | lmat22e12 32457 | . . . 4 β’ (π β (1π2) = π΅) |
36 | 34, 35 | oveq12d 7376 | . . 3 β’ (π β ((2π1) Β· (1π2)) = (πΆ Β· π΅)) |
37 | 33, 36 | oveq12d 7376 | . 2 β’ (π β (((1π1) Β· (2π2)) β ((2π1) Β· (1π2))) = ((π΄ Β· π·) β (πΆ Β· π΅))) |
38 | 30, 37 | eqtrd 2773 | 1 β’ (π β (π½βπ) = ((π΄ Β· π·) β (πΆ Β· π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {cpr 4589 βcfv 6497 (class class class)co 7358 1c1 11057 + caddc 11059 βcn 12158 2c2 12213 3c3 12214 β€cz 12504 ...cfz 13430 ..^cfzo 13573 β―chash 14236 Word cword 14408 β¨βcs2 14736 Basecbs 17088 .rcmulr 17139 -gcsg 18755 Ringcrg 19969 Mat cmat 21770 maDet cmdat 21949 litMatclmat 32449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-xnn0 12491 df-z 12505 df-dec 12624 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-fac 14180 df-bc 14209 df-hash 14237 df-word 14409 df-lsw 14457 df-concat 14465 df-s1 14490 df-substr 14535 df-pfx 14565 df-splice 14644 df-reverse 14653 df-s2 14743 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-efmnd 18684 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mulg 18878 df-subg 18930 df-ghm 19011 df-gim 19054 df-cntz 19102 df-oppg 19129 df-symg 19154 df-pmtr 19229 df-psgn 19278 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-rnghom 20153 df-subrg 20234 df-sra 20649 df-rgmod 20650 df-cnfld 20813 df-zring 20886 df-zrh 20920 df-dsmm 21154 df-frlm 21169 df-mat 21771 df-mdet 21950 df-lmat 32450 |
This theorem is referenced by: (None) |
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