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 11711 | . . . . 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 14233 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
8 | lmat22.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | lmat22.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
10 | 8, 9 | s2cld 14233 | . . . . 5 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
11 | 7, 10 | s2cld 14233 | . . . 4 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
12 | s2len 14251 | . . . . 5 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
14 | 2, 5, 6, 8, 9 | lmat22lem 31082 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
15 | lmat22det.v | . . . 4 ⊢ 𝑉 = (Base‘𝑅) | |
16 | eqid 2821 | . . . 4 ⊢ ((1...2) Mat 𝑅) = ((1...2) Mat 𝑅) | |
17 | eqid 2821 | . . . 4 ⊢ (Base‘((1...2) Mat 𝑅)) = (Base‘((1...2) Mat 𝑅)) | |
18 | 2, 4, 11, 13, 14, 15, 16, 17, 1 | lmatcl 31081 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) |
19 | 2z 12015 | . . . . . 6 ⊢ 2 ∈ ℤ | |
20 | fzval3 13107 | . . . . . 6 ⊢ (2 ∈ ℤ → (1...2) = (1..^(2 + 1))) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (1...2) = (1..^(2 + 1)) |
22 | 2p1e3 11780 | . . . . . 6 ⊢ (2 + 1) = 3 | |
23 | 22 | oveq2i 7167 | . . . . 5 ⊢ (1..^(2 + 1)) = (1..^3) |
24 | fzo13pr 13122 | . . . . 5 ⊢ (1..^3) = {1, 2} | |
25 | 21, 23, 24 | 3eqtri 2848 | . . . 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 21240 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
30 | 1, 18, 29 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
31 | 2, 5, 6, 8, 9 | lmat22e11 31083 | . . . 4 ⊢ (𝜑 → (1𝑀1) = 𝐴) |
32 | 2, 5, 6, 8, 9 | lmat22e22 31086 | . . . 4 ⊢ (𝜑 → (2𝑀2) = 𝐷) |
33 | 31, 32 | oveq12d 7174 | . . 3 ⊢ (𝜑 → ((1𝑀1) · (2𝑀2)) = (𝐴 · 𝐷)) |
34 | 2, 5, 6, 8, 9 | lmat22e21 31085 | . . . 4 ⊢ (𝜑 → (2𝑀1) = 𝐶) |
35 | 2, 5, 6, 8, 9 | lmat22e12 31084 | . . . 4 ⊢ (𝜑 → (1𝑀2) = 𝐵) |
36 | 34, 35 | oveq12d 7174 | . . 3 ⊢ (𝜑 → ((2𝑀1) · (1𝑀2)) = (𝐶 · 𝐵)) |
37 | 33, 36 | oveq12d 7174 | . 2 ⊢ (𝜑 → (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
38 | 30, 37 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {cpr 4569 ‘cfv 6355 (class class class)co 7156 1c1 10538 + caddc 10540 ℕcn 11638 2c2 11693 3c3 11694 ℤcz 11982 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Word cword 13862 〈“cs2 14203 Basecbs 16483 .rcmulr 16566 -gcsg 18105 Ringcrg 19297 Mat cmat 21016 maDet cmdat 21193 litMatclmat 31076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-word 13863 df-lsw 13915 df-concat 13923 df-s1 13950 df-substr 14003 df-pfx 14033 df-splice 14112 df-reverse 14121 df-s2 14210 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-efmnd 18034 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-gim 18399 df-cntz 18447 df-oppg 18474 df-symg 18496 df-pmtr 18570 df-psgn 18619 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-rnghom 19467 df-subrg 19533 df-sra 19944 df-rgmod 19945 df-cnfld 20546 df-zring 20618 df-zrh 20651 df-dsmm 20876 df-frlm 20891 df-mat 21017 df-mdet 21194 df-lmat 31077 |
This theorem is referenced by: (None) |
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