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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 12184 | . . . . 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 14717 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
8 | lmat22.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | lmat22.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
10 | 8, 9 | s2cld 14717 | . . . . 5 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
11 | 7, 10 | s2cld 14717 | . . . 4 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
12 | s2len 14735 | . . . . 5 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
14 | 2, 5, 6, 8, 9 | lmat22lem 32201 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
15 | lmat22det.v | . . . 4 ⊢ 𝑉 = (Base‘𝑅) | |
16 | eqid 2737 | . . . 4 ⊢ ((1...2) Mat 𝑅) = ((1...2) Mat 𝑅) | |
17 | eqid 2737 | . . . 4 ⊢ (Base‘((1...2) Mat 𝑅)) = (Base‘((1...2) Mat 𝑅)) | |
18 | 2, 4, 11, 13, 14, 15, 16, 17, 1 | lmatcl 32200 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) |
19 | 2z 12493 | . . . . . 6 ⊢ 2 ∈ ℤ | |
20 | fzval3 13595 | . . . . . 6 ⊢ (2 ∈ ℤ → (1...2) = (1..^(2 + 1))) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (1...2) = (1..^(2 + 1)) |
22 | 2p1e3 12253 | . . . . . 6 ⊢ (2 + 1) = 3 | |
23 | 22 | oveq2i 7362 | . . . . 5 ⊢ (1..^(2 + 1)) = (1..^3) |
24 | fzo13pr 13610 | . . . . 5 ⊢ (1..^3) = {1, 2} | |
25 | 21, 23, 24 | 3eqtri 2769 | . . . 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 21931 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
30 | 1, 18, 29 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
31 | 2, 5, 6, 8, 9 | lmat22e11 32202 | . . . 4 ⊢ (𝜑 → (1𝑀1) = 𝐴) |
32 | 2, 5, 6, 8, 9 | lmat22e22 32205 | . . . 4 ⊢ (𝜑 → (2𝑀2) = 𝐷) |
33 | 31, 32 | oveq12d 7369 | . . 3 ⊢ (𝜑 → ((1𝑀1) · (2𝑀2)) = (𝐴 · 𝐷)) |
34 | 2, 5, 6, 8, 9 | lmat22e21 32204 | . . . 4 ⊢ (𝜑 → (2𝑀1) = 𝐶) |
35 | 2, 5, 6, 8, 9 | lmat22e12 32203 | . . . 4 ⊢ (𝜑 → (1𝑀2) = 𝐵) |
36 | 34, 35 | oveq12d 7369 | . . 3 ⊢ (𝜑 → ((2𝑀1) · (1𝑀2)) = (𝐶 · 𝐵)) |
37 | 33, 36 | oveq12d 7369 | . 2 ⊢ (𝜑 → (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
38 | 30, 37 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {cpr 4586 ‘cfv 6493 (class class class)co 7351 1c1 11010 + caddc 11012 ℕcn 12111 2c2 12166 3c3 12167 ℤcz 12457 ...cfz 13378 ..^cfzo 13521 ♯chash 14183 Word cword 14355 〈“cs2 14687 Basecbs 17042 .rcmulr 17093 -gcsg 18709 Ringcrg 19917 Mat cmat 21705 maDet cmdat 21884 litMatclmat 32195 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-addf 11088 ax-mulf 11089 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-oadd 8408 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-dju 9795 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-xnn0 12444 df-z 12458 df-dec 12577 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-fac 14127 df-bc 14156 df-hash 14184 df-word 14356 df-lsw 14404 df-concat 14412 df-s1 14437 df-substr 14486 df-pfx 14516 df-splice 14595 df-reverse 14604 df-s2 14694 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-starv 17107 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-unif 17115 df-hom 17116 df-cco 17117 df-0g 17282 df-gsum 17283 df-prds 17288 df-pws 17290 df-mre 17425 df-mrc 17426 df-acs 17428 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mhm 18560 df-submnd 18561 df-efmnd 18638 df-grp 18710 df-minusg 18711 df-sbg 18712 df-mulg 18831 df-subg 18883 df-ghm 18964 df-gim 19007 df-cntz 19055 df-oppg 19082 df-symg 19107 df-pmtr 19182 df-psgn 19231 df-cmn 19522 df-abl 19523 df-mgp 19855 df-ur 19872 df-ring 19919 df-cring 19920 df-rnghom 20098 df-subrg 20172 df-sra 20585 df-rgmod 20586 df-cnfld 20749 df-zring 20822 df-zrh 20856 df-dsmm 21090 df-frlm 21105 df-mat 21706 df-mdet 21885 df-lmat 32196 |
This theorem is referenced by: (None) |
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