![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 11698 | . . . . 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 14224 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
8 | lmat22.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | lmat22.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
10 | 8, 9 | s2cld 14224 | . . . . 5 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
11 | 7, 10 | s2cld 14224 | . . . 4 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
12 | s2len 14242 | . . . . 5 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
14 | 2, 5, 6, 8, 9 | lmat22lem 31170 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
15 | lmat22det.v | . . . 4 ⊢ 𝑉 = (Base‘𝑅) | |
16 | eqid 2798 | . . . 4 ⊢ ((1...2) Mat 𝑅) = ((1...2) Mat 𝑅) | |
17 | eqid 2798 | . . . 4 ⊢ (Base‘((1...2) Mat 𝑅)) = (Base‘((1...2) Mat 𝑅)) | |
18 | 2, 4, 11, 13, 14, 15, 16, 17, 1 | lmatcl 31169 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) |
19 | 2z 12002 | . . . . . 6 ⊢ 2 ∈ ℤ | |
20 | fzval3 13101 | . . . . . 6 ⊢ (2 ∈ ℤ → (1...2) = (1..^(2 + 1))) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (1...2) = (1..^(2 + 1)) |
22 | 2p1e3 11767 | . . . . . 6 ⊢ (2 + 1) = 3 | |
23 | 22 | oveq2i 7146 | . . . . 5 ⊢ (1..^(2 + 1)) = (1..^3) |
24 | fzo13pr 13116 | . . . . 5 ⊢ (1..^3) = {1, 2} | |
25 | 21, 23, 24 | 3eqtri 2825 | . . . 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 21236 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
30 | 1, 18, 29 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
31 | 2, 5, 6, 8, 9 | lmat22e11 31171 | . . . 4 ⊢ (𝜑 → (1𝑀1) = 𝐴) |
32 | 2, 5, 6, 8, 9 | lmat22e22 31174 | . . . 4 ⊢ (𝜑 → (2𝑀2) = 𝐷) |
33 | 31, 32 | oveq12d 7153 | . . 3 ⊢ (𝜑 → ((1𝑀1) · (2𝑀2)) = (𝐴 · 𝐷)) |
34 | 2, 5, 6, 8, 9 | lmat22e21 31173 | . . . 4 ⊢ (𝜑 → (2𝑀1) = 𝐶) |
35 | 2, 5, 6, 8, 9 | lmat22e12 31172 | . . . 4 ⊢ (𝜑 → (1𝑀2) = 𝐵) |
36 | 34, 35 | oveq12d 7153 | . . 3 ⊢ (𝜑 → ((2𝑀1) · (1𝑀2)) = (𝐶 · 𝐵)) |
37 | 33, 36 | oveq12d 7153 | . 2 ⊢ (𝜑 → (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
38 | 30, 37 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {cpr 4527 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 ℕcn 11625 2c2 11680 3c3 11681 ℤcz 11969 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 〈“cs2 14194 Basecbs 16475 .rcmulr 16558 -gcsg 18097 Ringcrg 19290 Mat cmat 21012 maDet cmdat 21189 litMatclmat 31164 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-xor 1503 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-reverse 14112 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-gim 18391 df-cntz 18439 df-oppg 18466 df-symg 18488 df-pmtr 18562 df-psgn 18611 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-rnghom 19463 df-subrg 19526 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-dsmm 20421 df-frlm 20436 df-mat 21013 df-mdet 21190 df-lmat 31165 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |