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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 2622 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | mdetr0.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | mdetr0.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
6 | crngring 18558 | . . . . . 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 18569 | . . . . 5 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝐾) |
11 | 10 | 3ad2ant1 1082 | . . 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 20410 | . 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 18587 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐾) → ( 0 (.r‘𝑅) 0 ) = 0 ) |
16 | 7, 10, 15 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → ( 0 (.r‘𝑅) 0 ) = 0 ) |
17 | 16 | ifeq1d 4104 | . . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (.r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋)) |
18 | 17 | mpt2eq3dv 6721 | . . 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 6195 | . 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 2622 | . . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
21 | eqid 2622 | . . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
22 | 1, 20, 21, 2 | mdetf 20401 | . . . . 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 4131 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾) |
25 | 20, 2, 21, 5, 4, 24 | matbas2d 20229 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅))) |
26 | 23, 25 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾) |
27 | 2, 3, 8 | ringlz 18587 | . . 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 693 | . 2 ⊢ (𝜑 → ( 0 (.r‘𝑅)(𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)))) = 0 ) |
29 | 14, 19, 28 | 3eqtr3d 2664 | 1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ifcif 4086 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 Fincfn 7955 Basecbs 15857 .rcmulr 15942 0gc0g 16100 Ringcrg 18547 CRingccrg 18548 Mat cmat 20213 maDet cmdat 20390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-sra 19172 df-rgmod 19173 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-dsmm 20076 df-frlm 20091 df-mat 20214 df-mdet 20391 |
This theorem is referenced by: mdet0 20412 madugsum 20449 matunitlindflem1 33405 |
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