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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 2739 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | mdetr0.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | mdetr0.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
6 | crngring 19776 | . . . . . 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 19789 | . . . . 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 21734 | . 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 19807 | . . . . . 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 4483 | . . . 4 ⊢ (𝜑 → if(𝑖 = 𝐼, ( 0 (.r‘𝑅) 0 ), 𝑋) = if(𝑖 = 𝐼, 0 , 𝑋)) |
18 | 17 | mpoeq3dv 7345 | . . 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 6772 | . 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 2739 | . . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
21 | eqid 2739 | . . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
22 | 1, 20, 21, 2 | mdetf 21725 | . . . . 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 4510 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 0 , 𝑋) ∈ 𝐾) |
25 | 20, 2, 21, 5, 4, 24 | matbas2d 21553 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋)) ∈ (Base‘(𝑁 Mat 𝑅))) |
26 | 23, 25 | ffvelrnd 6956 | . . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) ∈ 𝐾) |
27 | 2, 3, 8 | ringlz 19807 | . . 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 2787 | 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 1541 ∈ wcel 2109 ifcif 4464 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 ∈ cmpo 7270 Fincfn 8707 Basecbs 16893 .rcmulr 16944 0gc0g 17131 Ringcrg 19764 CRingccrg 19765 Mat cmat 21535 maDet cmdat 21714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-sup 9162 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-xnn0 12289 df-z 12303 df-dec 12420 df-uz 12565 df-rp 12713 df-fz 13222 df-fzo 13365 df-seq 13703 df-exp 13764 df-hash 14026 df-word 14199 df-lsw 14247 df-concat 14255 df-s1 14282 df-substr 14335 df-pfx 14365 df-splice 14444 df-reverse 14453 df-s2 14542 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-0g 17133 df-gsum 17134 df-prds 17139 df-pws 17141 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mhm 18411 df-submnd 18412 df-efmnd 18489 df-grp 18561 df-minusg 18562 df-mulg 18682 df-subg 18733 df-ghm 18813 df-gim 18856 df-cntz 18904 df-oppg 18931 df-symg 18956 df-pmtr 19031 df-psgn 19080 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-cring 19767 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-rnghom 19940 df-drng 19974 df-subrg 20003 df-sra 20415 df-rgmod 20416 df-cnfld 20579 df-zring 20652 df-zrh 20686 df-dsmm 20920 df-frlm 20935 df-mat 21536 df-mdet 21715 |
This theorem is referenced by: mdet0 21736 madugsum 21773 matunitlindflem1 35752 |
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