mdet0 - Metamath Proof Explorer

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Theorem mdet0 21755

Description: The determinant of the zero matrix (of dimension greater 0!) is 0. (Contributed by AV, 17-Aug-2019.) (Revised by AV, 3-Jul-2022.)

**Hypotheses**
Ref	Expression
mdet0.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdet0.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdet0.z	⊢ 𝑍 = (0_g‘𝐴)
mdet0.0	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
mdet0	⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷‘𝑍) = 0 )

Proof of Theorem mdet0 Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4280	. . 3 ⊢ (𝑁 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝑁)
2		crngring 19795	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim1ci 616	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4	3	adantr 481	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5		mdet0.z	. . . . . . . . 9 ⊢ 𝑍 = (0_g‘𝐴)
6		mdet0.a	. . . . . . . . . 10 ⊢ 𝐴 = (𝑁 Mat 𝑅)
7		mdet0.0	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑅)
8	6, 7	mat0op 21568	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝐴) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
9	5, 8	eqtrid 2790	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑍 = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
10	4, 9	syl 17	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑍 = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
11	10	fveq2d 6778	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )))
12		ifid 4499	. . . . . . . . . 10 ⊢ if(𝑥 = 𝑖, 0 , 0 ) = 0
13	12	eqcomi 2747	. . . . . . . . 9 ⊢ 0 = if(𝑥 = 𝑖, 0 , 0 )
14	13	a1i 11	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 0 = if(𝑥 = 𝑖, 0 , 0 ))
15	14	mpoeq3dv 7354	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 )))
16	15	fveq2d 6778	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))))
17		mdet0.d	. . . . . . 7 ⊢ 𝐷 = (𝑁 maDet 𝑅)
18		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
19		simpll 764	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CRing)
20		simpr 485	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
21	20	adantr 481	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin)
22		ringmnd 19793	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
23	2, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Mnd)
24	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Mnd)
25	18, 7	mndidcl 18400	. . . . . . . . . 10 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
26	24, 25	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 0 ∈ (Base‘𝑅))
27	26	adantr 481	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
28	27	3ad2ant1 1132	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
29		simpr 485	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁)
30	17, 18, 7, 19, 21, 28, 29	mdetr0 21754	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))) = 0 )
31	11, 16, 30	3eqtrd 2782	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = 0 )
32	31	ex 413	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 ))
33	32	exlimdv 1936	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (∃𝑖 𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 ))
34	1, 33	syl5bi 241	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ≠ ∅ → (𝐷‘𝑍) = 0 ))
35	34	3impia 1116	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷‘𝑍) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 ifcif 4459 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 Fincfn 8733 Basecbs 16912 0_gc0g 17150 Mndcmnd 18385 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-sbg 18582 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-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-lmod 20125 df-lss 20194 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: (None)

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