mdet0 - Metamath Proof Explorer

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Theorem mdet0 22522

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 4303	. . 3 ⊢ (𝑁 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝑁)
2		crngring 20164	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim1ci 616	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4	3	adantr 480	. . . . . . . 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 22335	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝐴) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
9	5, 8	eqtrid 2778	. . . . . . . 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 6826	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )))
12		ifid 4516	. . . . . . . . . 10 ⊢ if(𝑥 = 𝑖, 0 , 0 ) = 0
13	12	eqcomi 2740	. . . . . . . . 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 7425	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 )))
16	15	fveq2d 6826	. . . . . 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 2731	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
19		simpll 766	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CRing)
20		simpr 484	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
21	20	adantr 480	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin)
22		ringmnd 20162	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
23	2, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Mnd)
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Mnd)
25	18, 7	mndidcl 18657	. . . . . . . . . 10 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
26	24, 25	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 0 ∈ (Base‘𝑅))
27	26	adantr 480	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
28	27	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
29		simpr 484	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁)
30	17, 18, 7, 19, 21, 28, 29	mdetr0 22521	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))) = 0 )
31	11, 16, 30	3eqtrd 2770	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = 0 )
32	31	ex 412	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 ))
33	32	exlimdv 1934	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (∃𝑖 𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 ))
34	1, 33	biimtrid 242	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ≠ ∅ → (𝐷‘𝑍) = 0 ))
35	34	3impia 1117	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷‘𝑍) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ≠ wne 2928 ∅c0 4283 ifcif 4475 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 Fincfn 8869 Basecbs 17120 0_gc0g 17343 Mndcmnd 18642 Ringcrg 20152 CRingccrg 20153 Mat cmat 22323 maDet cmdat 22500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1513 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-ot 4585 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14504 df-substr 14549 df-pfx 14579 df-splice 14657 df-reverse 14666 df-s2 14755 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-efmnd 18777 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19126 df-gim 19172 df-cntz 19230 df-oppg 19259 df-symg 19283 df-pmtr 19355 df-psgn 19404 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-ring 20154 df-cring 20155 df-oppr 20256 df-dvdsr 20276 df-unit 20277 df-invr 20307 df-dvr 20320 df-rhm 20391 df-subrng 20462 df-subrg 20486 df-drng 20647 df-lmod 20796 df-lss 20866 df-sra 21108 df-rgmod 21109 df-cnfld 21293 df-zring 21385 df-zrh 21441 df-dsmm 21670 df-frlm 21685 df-mat 22324 df-mdet 22501

This theorem is referenced by: (None)

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