mdet0 - Metamath Proof Explorer

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Theorem mdet0 20817

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 4158	. . 3 ⊢ (𝑁 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝑁)
2		crngring 18945	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim1i 608	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
4	3	ancomd 455	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5	4	adantr 474	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6		mdet0.z	. . . . . . . . 9 ⊢ 𝑍 = (0_g‘𝐴)
7		mdet0.a	. . . . . . . . . 10 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8		mdet0.0	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑅)
9	7, 8	mat0op 20629	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝐴) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
10	6, 9	syl5eq 2825	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑍 = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
11	5, 10	syl 17	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑍 = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))
12	11	fveq2d 6450	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )))
13		ifid 4345	. . . . . . . . . 10 ⊢ if(𝑥 = 𝑖, 0 , 0 ) = 0
14	13	eqcomi 2786	. . . . . . . . 9 ⊢ 0 = if(𝑥 = 𝑖, 0 , 0 )
15	14	a1i 11	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 0 = if(𝑥 = 𝑖, 0 , 0 ))
16	15	mpt2eq3dv 6998	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 )))
17	16	fveq2d 6450	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))))
18		mdet0.d	. . . . . . 7 ⊢ 𝐷 = (𝑁 maDet 𝑅)
19		eqid 2777	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
20		simpll 757	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CRing)
21		simpr 479	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
22	21	adantr 474	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin)
23		ringmnd 18943	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
24	2, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Mnd)
25	24	adantr 474	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Mnd)
26	19, 8	mndidcl 17694	. . . . . . . . . 10 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
27	25, 26	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 0 ∈ (Base‘𝑅))
28	27	adantr 474	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
29	28	3ad2ant1 1124	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
30		simpr 479	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁)
31	18, 19, 8, 20, 22, 29, 30	mdetr0 20816	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))) = 0 )
32	12, 17, 31	3eqtrd 2817	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = 0 )
33	32	ex 403	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 ))
34	33	exlimdv 1976	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (∃𝑖 𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 ))
35	1, 34	syl5bi 234	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ≠ ∅ → (𝐷‘𝑍) = 0 ))
36	35	3impia 1106	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷‘𝑍) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∃wex 1823 ∈ wcel 2106 ≠ wne 2968 ∅c0 4140 ifcif 4306 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 Fincfn 8241 Basecbs 16255 0_gc0g 16486 Mndcmnd 17680 Ringcrg 18934 CRingccrg 18935 Mat cmat 20617 maDet cmdat 20795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-xor 1583 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-ot 4406 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-xnn0 11715 df-z 11729 df-dec 11846 df-uz 11993 df-rp 12138 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-word 13600 df-lsw 13653 df-concat 13661 df-s1 13686 df-substr 13731 df-pfx 13780 df-splice 13887 df-reverse 13905 df-s2 13999 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-gim 18085 df-cntz 18133 df-oppg 18159 df-symg 18181 df-pmtr 18245 df-psgn 18294 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-rnghom 19104 df-drng 19141 df-subrg 19170 df-lmod 19257 df-lss 19325 df-sra 19569 df-rgmod 19570 df-cnfld 20143 df-zring 20215 df-zrh 20248 df-dsmm 20475 df-frlm 20490 df-mat 20618 df-mdet 20796

This theorem is referenced by: (None)

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