mdetero - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mdetero		Structured version Visualization version GIF version

Theorem mdetero 21759

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.)

**Hypotheses**
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)

**Assertion**
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)))))

Distinct variable groups: 𝜑,𝑖,𝑗 𝑖,𝐾,𝑗 𝑖,𝑁,𝑗 𝑖,𝐼,𝑗 𝑖,𝐽,𝑗 𝑖,𝑋 𝑖,𝑌 𝑖,𝑊,𝑗 · ,𝑖,𝑗 + ,𝑖,𝑗 Allowed substitution hints: 𝐷(𝑖,𝑗) 𝑅(𝑖,𝑗) 𝑋(𝑗) 𝑌(𝑗) 𝑍(𝑖,𝑗)

**Proof of Theorem mdetero**
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 1130	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
8		crngring 19795	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	4, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	9	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
11		mdetero.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐾)
12	11	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑊 ∈ 𝐾)
13		mdetero.y	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
14	13	3adant2 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
15		mdetero.t	. . . . 5 ⊢ · = (._r‘𝑅)
16	2, 15	ringcl 19800	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑊 · 𝑌) ∈ 𝐾)
17	10, 12, 14, 16	syl3anc 1370	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑊 · 𝑌) ∈ 𝐾)
18		mdetero.z	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)
19	14, 18	ifcld 4505	. . 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 21756	. 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 21753	. . . 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 2738	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘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 21758	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
27	26	oveq2d 7291	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍))))) = (𝑊 · (0_g‘𝑅)))
28	2, 15, 23	ringrz 19827	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾) → (𝑊 · (0_g‘𝑅)) = (0_g‘𝑅))
29	9, 11, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑊 · (0_g‘𝑅)) = (0_g‘𝑅))
30	22, 27, 29	3eqtrd 2782	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
31	30	oveq2d 7291	. 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)))) + (0_g‘𝑅)))
32		ringgrp 19788	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
33	9, 32	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
34		eqid 2738	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
35		eqid 2738	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
36	1, 34, 35, 2	mdetf 21744	. . . . 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 4505	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)) ∈ 𝐾)
39	34, 2, 35, 5, 4, 38	matbas2d 21572	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))) ∈ (Base‘(𝑁 Mat 𝑅)))
40	37, 39	ffvelrnd 6962	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) ∈ 𝐾)
41	2, 3, 23	grprid 18610	. . 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)))) + (0_g‘𝑅)) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))
42	33, 40, 41	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (0_g‘𝑅)) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))
43	21, 31, 42	3eqtrd 2782	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ifcif 4459 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 Fincfn 8733 Basecbs 16912 +_gcplusg 16962 ._rcmulr 16963 0_gc0g 17150 Grpcgrp 18577 Ringcrg 19783 CRingccrg 19784 Mat cmat 21554 maDet cmdat 21733

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-reverse 14472 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-efmnd 18508 df-grp 18580 df-minusg 18581 df-mulg 18701 df-subg 18752 df-ghm 18832 df-gim 18875 df-cntz 18923 df-oppg 18950 df-symg 18975 df-pmtr 19050 df-psgn 19099 df-evpm 19100 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-dsmm 20939 df-frlm 20954 df-mat 21555 df-mdet 21734

This theorem is referenced by: maducoeval2 21789 matunitlindflem1 35773

W3C validator