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Mirrors > Home > MPE Home > Th. List > mdetero | Structured version Visualization version GIF version |
Description: The determinant function is multilinear (additive and homogeneous for each row (matrices are given explicitly by their entries). Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 16-Jul-2018.) |
Ref | Expression |
---|---|
mdetero.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
mdetero.k | ⊢ 𝐾 = (Base‘𝑅) |
mdetero.p | ⊢ + = (+g‘𝑅) |
mdetero.t | ⊢ · = (.r‘𝑅) |
mdetero.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
mdetero.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mdetero.x | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) |
mdetero.y | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾) |
mdetero.z | ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾) |
mdetero.w | ⊢ (𝜑 → 𝑊 ∈ 𝐾) |
mdetero.i | ⊢ (𝜑 → 𝐼 ∈ 𝑁) |
mdetero.j | ⊢ (𝜑 → 𝐽 ∈ 𝑁) |
mdetero.ij | ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
Ref | Expression |
---|---|
mdetero | ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdetero.d | . . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
2 | mdetero.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
3 | mdetero.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | mdetero.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | mdetero.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
6 | mdetero.x | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) | |
7 | 6 | 3adant2 1073 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) |
8 | crngring 18381 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
9 | 4, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
10 | 9 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
11 | mdetero.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐾) | |
12 | 11 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑊 ∈ 𝐾) |
13 | mdetero.y | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾) | |
14 | 13 | 3adant2 1073 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾) |
15 | mdetero.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
16 | 2, 15 | ringcl 18384 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑊 · 𝑌) ∈ 𝐾) |
17 | 10, 12, 14, 16 | syl3anc 1318 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑊 · 𝑌) ∈ 𝐾) |
18 | mdetero.z | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾) | |
19 | 14, 18 | ifcld 4081 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐽, 𝑌, 𝑍) ∈ 𝐾) |
20 | mdetero.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
21 | 1, 2, 3, 4, 5, 7, 17, 19, 20 | mdetrlin2 20232 | . 2 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))))) |
22 | 1, 2, 15, 4, 5, 14, 19, 11, 20 | mdetrsca2 20229 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝑊 · (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍)))))) |
23 | eqid 2610 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
24 | mdetero.j | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝑁) | |
25 | mdetero.ij | . . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
26 | 1, 2, 23, 4, 5, 13, 18, 20, 24, 25 | mdetralt2 20234 | . . . . 5 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0g‘𝑅)) |
27 | 26 | oveq2d 6565 | . . . 4 ⊢ (𝜑 → (𝑊 · (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍))))) = (𝑊 · (0g‘𝑅))) |
28 | 2, 15, 23 | ringrz 18411 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾) → (𝑊 · (0g‘𝑅)) = (0g‘𝑅)) |
29 | 9, 11, 28 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝑊 · (0g‘𝑅)) = (0g‘𝑅)) |
30 | 22, 27, 29 | 3eqtrd 2648 | . . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0g‘𝑅)) |
31 | 30 | oveq2d 6565 | . 2 ⊢ (𝜑 → ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍))))) = ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (0g‘𝑅))) |
32 | ringgrp 18375 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
33 | 9, 32 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
34 | eqid 2610 | . . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
35 | eqid 2610 | . . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
36 | 1, 34, 35, 2 | mdetf 20220 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾) |
37 | 4, 36 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾) |
38 | 7, 19 | ifcld 4081 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)) ∈ 𝐾) |
39 | 34, 2, 35, 5, 4, 38 | matbas2d 20048 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))) ∈ (Base‘(𝑁 Mat 𝑅))) |
40 | 37, 39 | ffvelrnd 6268 | . . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) ∈ 𝐾) |
41 | 2, 3, 23 | grprid 17276 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) ∈ 𝐾) → ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (0g‘𝑅)) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))))) |
42 | 33, 40, 41 | syl2anc 691 | . 2 ⊢ (𝜑 → ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (0g‘𝑅)) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))))) |
43 | 21, 31, 42 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ifcif 4036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Fincfn 7841 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 0gc0g 15923 Grpcgrp 17245 Ringcrg 18370 CRingccrg 18371 Mat cmat 20032 maDet cmdat 20209 |
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-fal 1481 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-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-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-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-evpm 17735 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 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 |
This theorem is referenced by: maducoeval2 20265 matunitlindflem1 32575 |
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