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 11062 | . . . . 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 13466 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
8 | lmat22.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | lmat22.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
10 | 8, 9 | s2cld 13466 | . . . . 5 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
11 | 7, 10 | s2cld 13466 | . . . 4 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
12 | s2len 13484 | . . . . 5 ⊢ (#‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (#‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
14 | 2, 5, 6, 8, 9 | lmat22lem 29211 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
15 | lmat22det.v | . . . 4 ⊢ 𝑉 = (Base‘𝑅) | |
16 | eqid 2610 | . . . 4 ⊢ ((1...2) Mat 𝑅) = ((1...2) Mat 𝑅) | |
17 | eqid 2610 | . . . 4 ⊢ (Base‘((1...2) Mat 𝑅)) = (Base‘((1...2) Mat 𝑅)) | |
18 | 2, 4, 11, 13, 14, 15, 16, 17, 1 | lmatcl 29210 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) |
19 | 2z 11286 | . . . . . 6 ⊢ 2 ∈ ℤ | |
20 | fzval3 12404 | . . . . . 6 ⊢ (2 ∈ ℤ → (1...2) = (1..^(2 + 1))) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (1...2) = (1..^(2 + 1)) |
22 | 2p1e3 11028 | . . . . . 6 ⊢ (2 + 1) = 3 | |
23 | 22 | oveq2i 6560 | . . . . 5 ⊢ (1..^(2 + 1)) = (1..^3) |
24 | fzo13pr 12419 | . . . . 5 ⊢ (1..^3) = {1, 2} | |
25 | 21, 23, 24 | 3eqtri 2636 | . . . 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 20256 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (Base‘((1...2) Mat 𝑅))) → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
30 | 1, 18, 29 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐽‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2)))) |
31 | 2, 5, 6, 8, 9 | lmat22e11 29212 | . . . 4 ⊢ (𝜑 → (1𝑀1) = 𝐴) |
32 | 2, 5, 6, 8, 9 | lmat22e22 29215 | . . . 4 ⊢ (𝜑 → (2𝑀2) = 𝐷) |
33 | 31, 32 | oveq12d 6567 | . . 3 ⊢ (𝜑 → ((1𝑀1) · (2𝑀2)) = (𝐴 · 𝐷)) |
34 | 2, 5, 6, 8, 9 | lmat22e21 29214 | . . . 4 ⊢ (𝜑 → (2𝑀1) = 𝐶) |
35 | 2, 5, 6, 8, 9 | lmat22e12 29213 | . . . 4 ⊢ (𝜑 → (1𝑀2) = 𝐵) |
36 | 34, 35 | oveq12d 6567 | . . 3 ⊢ (𝜑 → ((2𝑀1) · (1𝑀2)) = (𝐶 · 𝐵)) |
37 | 33, 36 | oveq12d 6567 | . 2 ⊢ (𝜑 → (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
38 | 30, 37 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {cpr 4127 ‘cfv 5804 (class class class)co 6549 1c1 9816 + caddc 9818 ℕcn 10897 2c2 10947 3c3 10948 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 〈“cs2 13437 Basecbs 15695 .rcmulr 15769 -gcsg 17247 Ringcrg 18370 Mat cmat 20032 maDet cmdat 20209 litMatclmat 29205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-xor 1457 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-reverse 13160 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-gim 17524 df-cntz 17573 df-oppg 17599 df-symg 17621 df-pmtr 17685 df-psgn 17734 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-rnghom 18538 df-subrg 18601 df-sra 18993 df-rgmod 18994 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-dsmm 19895 df-frlm 19910 df-mat 20033 df-mdet 20210 df-lmat 29206 |
This theorem is referenced by: (None) |
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