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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 2733 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | mdetr0.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | mdetr0.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
6 | crngring 19984 | . . . . . 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 19998 | . . . . 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 21976 | . 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 20019 | . . . . . 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 4509 | . . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (.r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋)) |
18 | 17 | mpoeq3dv 7440 | . . 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 6850 | . 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 2733 | . . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
21 | eqid 2733 | . . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
22 | 1, 20, 21, 2 | mdetf 21967 | . . . . 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 4536 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾) |
25 | 20, 2, 21, 5, 4, 24 | matbas2d 21795 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅))) |
26 | 23, 25 | ffvelcdmd 7040 | . . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾) |
27 | 2, 3, 8 | ringlz 20019 | . . 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 2781 | 1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ifcif 4490 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ∈ cmpo 7363 Fincfn 8889 Basecbs 17091 .rcmulr 17142 0gc0g 17329 Ringcrg 19972 CRingccrg 19973 Mat cmat 21777 maDet cmdat 21956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-word 14412 df-lsw 14460 df-concat 14468 df-s1 14493 df-substr 14538 df-pfx 14568 df-splice 14647 df-reverse 14656 df-s2 14746 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-0g 17331 df-gsum 17332 df-prds 17337 df-pws 17339 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-efmnd 18687 df-grp 18759 df-minusg 18760 df-mulg 18881 df-subg 18933 df-ghm 19014 df-gim 19057 df-cntz 19105 df-oppg 19132 df-symg 19157 df-pmtr 19232 df-psgn 19281 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-rnghom 20156 df-drng 20221 df-subrg 20262 df-sra 20678 df-rgmod 20679 df-cnfld 20820 df-zring 20893 df-zrh 20927 df-dsmm 21161 df-frlm 21176 df-mat 21778 df-mdet 21957 |
This theorem is referenced by: mdet0 21978 madugsum 22015 matunitlindflem1 36124 |
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