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Mirrors > Home > MPE Home > Th. List > mdetuni | Structured version Visualization version GIF version |
Description: According to the definition in [Weierstrass] p. 272, the determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring. So for any multilinear (mdetuni.li and mdetuni.sc), alternating (mdetuni.al) and normalized (mdetuni.no) function D (mdetuni.ff) from the algebra of square matrices (mdetuni.a) to their underlying commutative ring (mdetuni.cr), the function value of this function D for a matrix F (mdetuni.f) is the determinant of this matrix. (Contributed by Stefan O'Rear, 15-Jul-2018.) (Revised by Alexander van der Vekens, 8-Feb-2019.) |
Ref | Expression |
---|---|
mdetuni.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mdetuni.b | ⊢ 𝐵 = (Base‘𝐴) |
mdetuni.k | ⊢ 𝐾 = (Base‘𝑅) |
mdetuni.0g | ⊢ 0 = (0g‘𝑅) |
mdetuni.1r | ⊢ 1 = (1r‘𝑅) |
mdetuni.pg | ⊢ + = (+g‘𝑅) |
mdetuni.tg | ⊢ · = (.r‘𝑅) |
mdetuni.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mdetuni.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mdetuni.ff | ⊢ (𝜑 → 𝐷:𝐵⟶𝐾) |
mdetuni.al | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 )) |
mdetuni.li | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) |
mdetuni.sc | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) |
mdetuni.e | ⊢ 𝐸 = (𝑁 maDet 𝑅) |
mdetuni.cr | ⊢ (𝜑 → 𝑅 ∈ CRing) |
mdetuni.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
mdetuni.no | ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 1 ) |
Ref | Expression |
---|---|
mdetuni | ⊢ (𝜑 → (𝐷‘𝐹) = (𝐸‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdetuni.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | mdetuni.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
3 | mdetuni.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
4 | mdetuni.0g | . . 3 ⊢ 0 = (0g‘𝑅) | |
5 | mdetuni.1r | . . 3 ⊢ 1 = (1r‘𝑅) | |
6 | mdetuni.pg | . . 3 ⊢ + = (+g‘𝑅) | |
7 | mdetuni.tg | . . 3 ⊢ · = (.r‘𝑅) | |
8 | mdetuni.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
9 | mdetuni.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
10 | mdetuni.ff | . . 3 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾) | |
11 | mdetuni.al | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 )) | |
12 | mdetuni.li | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) | |
13 | mdetuni.sc | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) | |
14 | mdetuni.e | . . 3 ⊢ 𝐸 = (𝑁 maDet 𝑅) | |
15 | mdetuni.cr | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
16 | mdetuni.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 | mdetuni0 22105 | . 2 ⊢ (𝜑 → (𝐷‘𝐹) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹))) |
18 | mdetuni.no | . . 3 ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 1 ) | |
19 | 18 | oveq1d 7419 | . 2 ⊢ (𝜑 → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)) = ( 1 · (𝐸‘𝐹))) |
20 | 14, 1, 2, 3 | mdetcl 22080 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ 𝐵) → (𝐸‘𝐹) ∈ 𝐾) |
21 | 15, 16, 20 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝐸‘𝐹) ∈ 𝐾) |
22 | 3, 7, 5 | ringlidm 20076 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐸‘𝐹) ∈ 𝐾) → ( 1 · (𝐸‘𝐹)) = (𝐸‘𝐹)) |
23 | 9, 21, 22 | syl2anc 585 | . 2 ⊢ (𝜑 → ( 1 · (𝐸‘𝐹)) = (𝐸‘𝐹)) |
24 | 17, 19, 23 | 3eqtrd 2777 | 1 ⊢ (𝜑 → (𝐷‘𝐹) = (𝐸‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∖ cdif 3944 {csn 4627 × cxp 5673 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ∘f cof 7663 Fincfn 8935 Basecbs 17140 +gcplusg 17193 .rcmulr 17194 0gc0g 17381 1rcur 19996 Ringcrg 20047 CRingccrg 20048 Mat cmat 21889 maDet cmdat 22068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-word 14461 df-lsw 14509 df-concat 14517 df-s1 14542 df-substr 14587 df-pfx 14617 df-splice 14696 df-reverse 14705 df-s2 14795 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-efmnd 18746 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-gim 19127 df-cntz 19175 df-oppg 19203 df-symg 19228 df-pmtr 19303 df-psgn 19352 df-evpm 19353 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-srg 20001 df-ring 20049 df-cring 20050 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-rnghom 20240 df-drng 20306 df-subrg 20349 df-lmod 20461 df-lss 20531 df-sra 20773 df-rgmod 20774 df-cnfld 20930 df-zring 21003 df-zrh 21037 df-dsmm 21271 df-frlm 21286 df-mamu 21868 df-mat 21890 df-mdet 22069 |
This theorem is referenced by: (None) |
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