![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mdet0 | Structured version Visualization version GIF version |
Description: The determinant of the zero matrix (of dimension greater 0!) is 0. (Contributed by AV, 17-Aug-2019.) (Revised by AV, 3-Jul-2022.) |
Ref | Expression |
---|---|
mdet0.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
mdet0.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mdet0.z | ⊢ 𝑍 = (0g‘𝐴) |
mdet0.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
mdet0 | ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷‘𝑍) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4347 | . . 3 ⊢ (𝑁 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝑁) | |
2 | crngring 20185 | . . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
3 | 2 | anim1ci 615 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
4 | 3 | adantr 480 | . . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
5 | mdet0.z | . . . . . . . . 9 ⊢ 𝑍 = (0g‘𝐴) | |
6 | mdet0.a | . . . . . . . . . 10 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
7 | mdet0.0 | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑅) | |
8 | 6, 7 | mat0op 22334 | . . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )) |
9 | 5, 8 | eqtrid 2780 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑍 = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )) |
10 | 4, 9 | syl 17 | . . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑍 = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )) |
11 | 10 | fveq2d 6901 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ))) |
12 | ifid 4569 | . . . . . . . . . 10 ⊢ if(𝑥 = 𝑖, 0 , 0 ) = 0 | |
13 | 12 | eqcomi 2737 | . . . . . . . . 9 ⊢ 0 = if(𝑥 = 𝑖, 0 , 0 ) |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 0 = if(𝑥 = 𝑖, 0 , 0 )) |
15 | 14 | mpoeq3dv 7499 | . . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 ) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))) |
16 | 15 | fveq2d 6901 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 0 )) = (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 )))) |
17 | mdet0.d | . . . . . . 7 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
18 | eqid 2728 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
19 | simpll 766 | . . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CRing) | |
20 | simpr 484 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin) | |
21 | 20 | adantr 480 | . . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin) |
22 | ringmnd 20183 | . . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
23 | 2, 22 | syl 17 | . . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Mnd) |
24 | 23 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Mnd) |
25 | 18, 7 | mndidcl 18709 | . . . . . . . . . 10 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
26 | 24, 25 | syl 17 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 0 ∈ (Base‘𝑅)) |
27 | 26 | adantr 480 | . . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 0 ∈ (Base‘𝑅)) |
28 | 27 | 3ad2ant1 1131 | . . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 0 ∈ (Base‘𝑅)) |
29 | simpr 484 | . . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁) | |
30 | 17, 18, 7, 19, 21, 28, 29 | mdetr0 22520 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑖, 0 , 0 ))) = 0 ) |
31 | 11, 16, 30 | 3eqtrd 2772 | . . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑖 ∈ 𝑁) → (𝐷‘𝑍) = 0 ) |
32 | 31 | ex 412 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 )) |
33 | 32 | exlimdv 1929 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (∃𝑖 𝑖 ∈ 𝑁 → (𝐷‘𝑍) = 0 )) |
34 | 1, 33 | biimtrid 241 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝑁 ≠ ∅ → (𝐷‘𝑍) = 0 )) |
35 | 34 | 3impia 1115 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷‘𝑍) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 ifcif 4529 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 Fincfn 8964 Basecbs 17180 0gc0g 17421 Mndcmnd 18694 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-sbg 18895 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-lmod 20745 df-lss 20816 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: (None) |
Copyright terms: Public domain | W3C validator |