mdetr0 - Metamath Proof Explorer

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Theorem mdetr0 22518

Description: The determinant of a matrix with a row containing only 0's is 0. (Contributed by SO, 16-Jul-2018.)

**Hypotheses**
Ref	Expression
mdetr0.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetr0.k	⊢ 𝐾 = (Base‘𝑅)
mdetr0.z	⊢ 0 = (0_g‘𝑅)
mdetr0.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mdetr0.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetr0.x	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
mdetr0.i	⊢ (𝜑 → 𝐼 ∈ 𝑁)

**Assertion**
Ref	Expression
mdetr0	⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )

Distinct variable groups: 𝜑,𝑖,𝑗 𝑖,𝐾,𝑗 𝑖,𝑁,𝑗 𝑖,𝐼,𝑗 0 ,𝑖,𝑗 𝑅,𝑖,𝑗 Allowed substitution hints: 𝐷(𝑖,𝑗) 𝑋(𝑖,𝑗)

**Proof of Theorem mdetr0**
Step	Hyp	Ref	Expression
1		mdetr0.d	. . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅)
2		mdetr0.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
3		eqid 2731	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
4		mdetr0.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
5		mdetr0.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
6		crngring 20161	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
7	4, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
8		mdetr0.z	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
9	2, 8	ring0cl 20183	. . . . 5 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
10	7, 9	syl 17	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐾)
11	10	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 0 ∈ 𝐾)
12		mdetr0.x	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
13		mdetr0.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑁)
14	1, 2, 3, 4, 5, 11, 12, 10, 13	mdetrsca2 22517	. 2 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, ( 0 (._r‘𝑅) 0 ), 𝑋))) = ( 0 (._r‘𝑅)(𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))))
15	2, 3, 8	ringlz 20209	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐾) → ( 0 (._r‘𝑅) 0 ) = 0 )
16	7, 10, 15	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( 0 (._r‘𝑅) 0 ) = 0 )
17	16	ifeq1d 4495	. . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (._r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋))
18	17	mpoeq3dv 7425	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, ( 0 (._r‘𝑅) 0 ), 𝑋)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))
19	18	fveq2d 6826	. 2 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, ( 0 (._r‘𝑅) 0 ), 𝑋))) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))))
20		eqid 2731	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
21		eqid 2731	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
22	1, 20, 21, 2	mdetf 22508	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾)
23	4, 22	syl 17	. . . 4 ⊢ (𝜑 → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾)
24	11, 12	ifcld 4522	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾)
25	20, 2, 21, 5, 4, 24	matbas2d 22336	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅)))
26	23, 25	ffvelcdmd 7018	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾)
27	2, 3, 8	ringlz 20209	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾) → ( 0 (._r‘𝑅)(𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))) = 0 )
28	7, 26, 27	syl2anc 584	. 2 ⊢ (𝜑 → ( 0 (._r‘𝑅)(𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))) = 0 )
29	14, 19, 28	3eqtr3d 2774	1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ifcif 4475 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 Fincfn 8869 Basecbs 17117 ._rcmulr 17159 0_gc0g 17340 Ringcrg 20149 CRingccrg 20150 Mat cmat 22320 maDet cmdat 22497

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 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-addf 11082 ax-mulf 11083

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 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-xnn0 12452 df-z 12466 df-dec 12586 df-uz 12730 df-rp 12888 df-fz 13405 df-fzo 13552 df-seq 13906 df-exp 13966 df-hash 14235 df-word 14418 df-lsw 14467 df-concat 14475 df-s1 14501 df-substr 14546 df-pfx 14576 df-splice 14654 df-reverse 14663 df-s2 14752 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-0g 17342 df-gsum 17343 df-prds 17348 df-pws 17350 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-mhm 18688 df-submnd 18689 df-efmnd 18774 df-grp 18846 df-minusg 18847 df-mulg 18978 df-subg 19033 df-ghm 19123 df-gim 19169 df-cntz 19227 df-oppg 19256 df-symg 19280 df-pmtr 19352 df-psgn 19401 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20459 df-subrg 20483 df-drng 20644 df-sra 21105 df-rgmod 21106 df-cnfld 21290 df-zring 21382 df-zrh 21438 df-dsmm 21667 df-frlm 21682 df-mat 22321 df-mdet 22498

This theorem is referenced by: mdet0 22519 madugsum 22556 matunitlindflem1 37655

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