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Mirrors > Home > MPE Home > Th. List > mdetr0 | Structured version Visualization version GIF version |
Description: The determinant of a matrix with a row containing only 0's is 0. (Contributed by SO, 16-Jul-2018.) |
Ref | Expression |
---|---|
mdetr0.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
mdetr0.k | ⊢ 𝐾 = (Base‘𝑅) |
mdetr0.z | ⊢ 0 = (0g‘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) |
Ref | Expression |
---|---|
mdetr0 | ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdetr0.d | . . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
2 | mdetr0.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
3 | eqid 2821 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | mdetr0.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | mdetr0.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
6 | crngring 19307 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
8 | mdetr0.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
9 | 2, 8 | ring0cl 19318 | . . . . 5 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝐾) |
11 | 10 | 3ad2ant1 1129 | . . 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 21212 | . 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 19336 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐾) → ( 0 (.r‘𝑅) 0 ) = 0 ) |
16 | 7, 10, 15 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ( 0 (.r‘𝑅) 0 ) = 0 ) |
17 | 16 | ifeq1d 4484 | . . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (.r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋)) |
18 | 17 | mpoeq3dv 7232 | . . 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 6673 | . 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 2821 | . . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
21 | eqid 2821 | . . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
22 | 1, 20, 21, 2 | mdetf 21203 | . . . . 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 4511 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾) |
25 | 20, 2, 21, 5, 4, 24 | matbas2d 21031 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅))) |
26 | 23, 25 | ffvelrnd 6851 | . . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾) |
27 | 2, 3, 8 | ringlz 19336 | . . 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 586 | . 2 ⊢ (𝜑 → ( 0 (.r‘𝑅)(𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))) = 0 ) |
29 | 14, 19, 28 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ifcif 4466 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 Fincfn 8508 Basecbs 16482 .rcmulr 16565 0gc0g 16712 Ringcrg 19296 CRingccrg 19297 Mat cmat 21015 maDet cmdat 21192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1501 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-xnn0 11967 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-word 13861 df-lsw 13914 df-concat 13922 df-s1 13949 df-substr 14002 df-pfx 14032 df-splice 14111 df-reverse 14120 df-s2 14209 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-efmnd 18033 df-grp 18105 df-minusg 18106 df-mulg 18224 df-subg 18275 df-ghm 18355 df-gim 18398 df-cntz 18446 df-oppg 18473 df-symg 18495 df-pmtr 18569 df-psgn 18618 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-rnghom 19466 df-drng 19503 df-subrg 19532 df-sra 19943 df-rgmod 19944 df-cnfld 20545 df-zring 20617 df-zrh 20650 df-dsmm 20875 df-frlm 20890 df-mat 21016 df-mdet 21193 |
This theorem is referenced by: mdet0 21214 madugsum 21251 matunitlindflem1 34887 |
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