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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 2728 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | mdetr0.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | mdetr0.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
6 | crngring 20185 | . . . . . 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 20203 | . . . . 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 22519 | . 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 20229 | . . . . . 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 4548 | . . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (.r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋)) |
18 | 17 | mpoeq3dv 7499 | . . 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 6901 | . 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 2728 | . . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
21 | eqid 2728 | . . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
22 | 1, 20, 21, 2 | mdetf 22510 | . . . . 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 4575 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾) |
25 | 20, 2, 21, 5, 4, 24 | matbas2d 22338 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅))) |
26 | 23, 25 | ffvelcdmd 7095 | . . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾) |
27 | 2, 3, 8 | ringlz 20229 | . . 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 2776 | 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 1534 ∈ wcel 2099 ifcif 4529 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 Fincfn 8964 Basecbs 17180 .rcmulr 17234 0gc0g 17421 Ringcrg 20173 CRingccrg 20174 Mat cmat 22320 maDet cmdat 22499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-xnn0 12576 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-word 14498 df-lsw 14546 df-concat 14554 df-s1 14579 df-substr 14624 df-pfx 14654 df-splice 14733 df-reverse 14742 df-s2 14832 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-efmnd 18821 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-ghm 19168 df-gim 19213 df-cntz 19268 df-oppg 19297 df-symg 19322 df-pmtr 19397 df-psgn 19446 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-drng 20626 df-sra 21058 df-rgmod 21059 df-cnfld 21280 df-zring 21373 df-zrh 21429 df-dsmm 21666 df-frlm 21681 df-mat 22321 df-mdet 22500 |
This theorem is referenced by: mdet0 22521 madugsum 22558 matunitlindflem1 37089 |
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