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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 | ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) |
lmat22.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
lmat22.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
lmat22.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
lmat22.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
lmat22det.t | ⊢ · = (.r‘𝑅) |
lmat22det.s | ⊢ − = (-g‘𝑅) |
lmat22det.v | ⊢ 𝑉 = (Base‘𝑅) |
lmat22det.j | ⊢ 𝐽 = ((1...2) maDet 𝑅) |
lmat22det.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
lmat22det | ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22det.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | lmat22.m | . . . 4 ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) | |
3 | 2nn 11431 | . . . . 5 ⊢ 2 ∈ ℕ | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℕ) |
5 | lmat22.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | lmat22.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
7 | 5, 6 | s2cld 13999 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
8 | lmat22.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | lmat22.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
10 | 8, 9 | s2cld 13999 | . . . . 5 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
11 | 7, 10 | s2cld 13999 | . . . 4 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
12 | s2len 14017 | . . . . 5 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
14 | 2, 5, 6, 8, 9 | lmat22lem 30424 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
15 | lmat22det.v | . . . 4 ⊢ 𝑉 = (Base‘𝑅) | |
16 | eqid 2825 | . . . 4 ⊢ ((1...2) Mat 𝑅) = ((1...2) Mat 𝑅) | |
17 | eqid 2825 | . . . 4 ⊢ (Base‘((1...2) Mat 𝑅)) = (Base‘((1...2) Mat 𝑅)) | |
18 | 2, 4, 11, 13, 14, 15, 16, 17, 1 | lmatcl 30423 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) |
19 | 2z 11744 | . . . . . 6 ⊢ 2 ∈ ℤ | |
20 | fzval3 12839 | . . . . . 6 ⊢ (2 ∈ ℤ → (1...2) = (1..^(2 + 1))) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (1...2) = (1..^(2 + 1)) |
22 | 2p1e3 11507 | . . . . . 6 ⊢ (2 + 1) = 3 | |
23 | 22 | oveq2i 6921 | . . . . 5 ⊢ (1..^(2 + 1)) = (1..^3) |
24 | fzo13pr 12854 | . . . . 5 ⊢ (1..^3) = {1, 2} | |
25 | 21, 23, 24 | 3eqtri 2853 | . . . 4 ⊢ (1...2) = {1, 2} |
26 | lmat22det.j | . . . 4 ⊢ 𝐽 = ((1...2) maDet 𝑅) | |
27 | lmat22det.s | . . . 4 ⊢ − = (-g‘𝑅) | |
28 | lmat22det.t | . . . 4 ⊢ · = (.r‘𝑅) | |
29 | 25, 26, 16, 17, 27, 28 | m2detleib 20812 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
30 | 1, 18, 29 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
31 | 2, 5, 6, 8, 9 | lmat22e11 30425 | . . . 4 ⊢ (𝜑 → (1𝑀1) = 𝐴) |
32 | 2, 5, 6, 8, 9 | lmat22e22 30428 | . . . 4 ⊢ (𝜑 → (2𝑀2) = 𝐷) |
33 | 31, 32 | oveq12d 6928 | . . 3 ⊢ (𝜑 → ((1𝑀1) · (2𝑀2)) = (𝐴 · 𝐷)) |
34 | 2, 5, 6, 8, 9 | lmat22e21 30427 | . . . 4 ⊢ (𝜑 → (2𝑀1) = 𝐶) |
35 | 2, 5, 6, 8, 9 | lmat22e12 30426 | . . . 4 ⊢ (𝜑 → (1𝑀2) = 𝐵) |
36 | 34, 35 | oveq12d 6928 | . . 3 ⊢ (𝜑 → ((2𝑀1) · (1𝑀2)) = (𝐶 · 𝐵)) |
37 | 33, 36 | oveq12d 6928 | . 2 ⊢ (𝜑 → (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
38 | 30, 37 | eqtrd 2861 | 1 ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 {cpr 4401 ‘cfv 6127 (class class class)co 6910 1c1 10260 + caddc 10262 ℕcn 11357 2c2 11413 3c3 11414 ℤcz 11711 ...cfz 12626 ..^cfzo 12767 ♯chash 13417 Word cword 13581 〈“cs2 13969 Basecbs 16229 .rcmulr 16313 -gcsg 17785 Ringcrg 18908 Mat cmat 20587 maDet cmdat 20765 litMatclmat 30418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-xor 1638 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-xnn0 11698 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-fac 13361 df-bc 13390 df-hash 13418 df-word 13582 df-lsw 13630 df-concat 13638 df-s1 13663 df-substr 13708 df-pfx 13757 df-splice 13864 df-reverse 13882 df-s2 13976 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-gim 18059 df-cntz 18107 df-oppg 18133 df-symg 18155 df-pmtr 18219 df-psgn 18268 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-rnghom 19078 df-subrg 19141 df-sra 19540 df-rgmod 19541 df-cnfld 20114 df-zring 20186 df-zrh 20219 df-dsmm 20446 df-frlm 20461 df-mat 20588 df-mdet 20766 df-lmat 30419 |
This theorem is referenced by: (None) |
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