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Theorem mdetr0 21662

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 2738	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
4		mdetr0.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
5		mdetr0.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
6		crngring 19710	. . . . . 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 19723	. . . . 5 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
10	7, 9	syl 17	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐾)
11	10	3ad2ant1 1131	. . 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 21661	. 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 19741	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐾) → ( 0 (._r‘𝑅) 0 ) = 0 )
16	7, 10, 15	syl2anc 583	. . . . 5 ⊢ (𝜑 → ( 0 (._r‘𝑅) 0 ) = 0 )
17	16	ifeq1d 4475	. . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (._r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋))
18	17	mpoeq3dv 7332	. . 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 6760	. 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 2738	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
21		eqid 2738	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
22	1, 20, 21, 2	mdetf 21652	. . . . 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 4502	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾)
25	20, 2, 21, 5, 4, 24	matbas2d 21480	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅)))
26	23, 25	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾)
27	2, 3, 8	ringlz 19741	. . 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 583	. 2 ⊢ (𝜑 → ( 0 (._r‘𝑅)(𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))) = 0 )
29	14, 19, 28	3eqtr3d 2786	1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ifcif 4456 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 Fincfn 8691 Basecbs 16840 ._rcmulr 16889 0_gc0g 17067 Ringcrg 19698 CRingccrg 19699 Mat cmat 21464 maDet cmdat 21641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-splice 14391 df-reverse 14400 df-s2 14489 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-efmnd 18423 df-grp 18495 df-minusg 18496 df-mulg 18616 df-subg 18667 df-ghm 18747 df-gim 18790 df-cntz 18838 df-oppg 18865 df-symg 18890 df-pmtr 18965 df-psgn 19014 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-dsmm 20849 df-frlm 20864 df-mat 21465 df-mdet 21642

This theorem is referenced by: mdet0 21663 madugsum 21700 matunitlindflem1 35700

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