mdet0 - Metamath Proof Explorer

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Theorem mdet0 22099

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 4345	. . 3 ⊢ (𝑁 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝑁)
2		crngring 20061	. . . . . . . . . 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 21912	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝐴) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
9	5, 8	eqtrid 2784	. . . . . . . 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 6892	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )))
12		ifid 4567	. . . . . . . . . 10 ⊢ if(𝑥 = 𝑖, 0 , 0 ) = 0
13	12	eqcomi 2741	. . . . . . . . 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 7484	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 )))
16	15	fveq2d 6892	. . . . . 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 2732	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
19		simpll 765	. . . . . . 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 20059	. . . . . . . . . . . 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 18636	. . . . . . . . . 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 1133	. . . . . . 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 22098	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))) = 0 )
31	11, 16, 30	3eqtrd 2776	. . . . 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	biimtrid 241	. 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 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∅c0 4321 ifcif 4527 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 Fincfn 8935 Basecbs 17140 0_gc0g 17381 Mndcmnd 18621 Ringcrg 20049 CRingccrg 20050 Mat cmat 21898 maDet cmdat 22077

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 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 19204 df-symg 19229 df-pmtr 19304 df-psgn 19353 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-rnghom 20243 df-drng 20309 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-cnfld 20937 df-zring 21010 df-zrh 21044 df-dsmm 21278 df-frlm 21293 df-mat 21899 df-mdet 22078

This theorem is referenced by: (None)

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