mdetr0 - Metamath Proof Explorer

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Theorem mdetr0 22570

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 2736	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
4		mdetr0.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
5		mdetr0.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
6		crngring 20226	. . . . . 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 20248	. . . . 5 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
10	7, 9	syl 17	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐾)
11	10	3ad2ant1 1134	. . 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 22569	. 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 20274	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐾) → ( 0 (._r‘𝑅) 0 ) = 0 )
16	7, 10, 15	syl2anc 585	. . . . 5 ⊢ (𝜑 → ( 0 (._r‘𝑅) 0 ) = 0 )
17	16	ifeq1d 4486	. . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (._r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋))
18	17	mpoeq3dv 7446	. . 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 6844	. 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 2736	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
21		eqid 2736	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
22	1, 20, 21, 2	mdetf 22560	. . . . 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 4513	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾)
25	20, 2, 21, 5, 4, 24	matbas2d 22388	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅)))
26	23, 25	ffvelcdmd 7037	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾)
27	2, 3, 8	ringlz 20274	. . 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 585	. 2 ⊢ (𝜑 → ( 0 (._r‘𝑅)(𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))) = 0 )
29	14, 19, 28	3eqtr3d 2779	1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ifcif 4466 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 Fincfn 8893 Basecbs 17179 ._rcmulr 17221 0_gc0g 17402 Ringcrg 20214 CRingccrg 20215 Mat cmat 22372 maDet cmdat 22549

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-word 14476 df-lsw 14525 df-concat 14533 df-s1 14559 df-substr 14604 df-pfx 14634 df-splice 14712 df-reverse 14721 df-s2 14810 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-efmnd 18837 df-grp 18912 df-minusg 18913 df-mulg 19044 df-subg 19099 df-ghm 19188 df-gim 19234 df-cntz 19292 df-oppg 19321 df-symg 19345 df-pmtr 19417 df-psgn 19466 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-drng 20708 df-sra 21168 df-rgmod 21169 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-dsmm 21712 df-frlm 21727 df-mat 22373 df-mdet 22550

This theorem is referenced by: mdet0 22571 madugsum 22608 matunitlindflem1 37937

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