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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smatcl | Structured version Visualization version GIF version |
Description: Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indices in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.) |
Ref | Expression |
---|---|
smatcl.a | ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) |
smatcl.b | ⊢ 𝐵 = (Base‘𝐴) |
smatcl.c | ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅)) |
smatcl.s | ⊢ 𝑆 = (𝐾(subMat1‘𝑀)𝐿) |
smatcl.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
smatcl.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) |
smatcl.l | ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) |
smatcl.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
Ref | Expression |
---|---|
smatcl | ⊢ (𝜑 → 𝑆 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smatcl.s | . . . 4 ⊢ 𝑆 = (𝐾(subMat1‘𝑀)𝐿) | |
2 | smatcl.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | smatcl.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
4 | smatcl.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) | |
5 | smatcl.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
6 | smatcl.a | . . . . . 6 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) | |
7 | eqid 2825 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | smatcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
9 | 6, 7, 8 | matbas2i 20602 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁)))) |
10 | 5, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁)))) |
11 | 1, 2, 2, 3, 4, 10 | smatrcl 30403 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))) |
12 | fzfi 13073 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
13 | 6, 8 | matrcl 20592 | . . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → ((1...𝑁) ∈ Fin ∧ 𝑅 ∈ V)) |
14 | 13 | simprd 491 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑅 ∈ V) |
15 | 5, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
16 | eqid 2825 | . . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅) | |
17 | 16, 7 | matbas2 20601 | . . . . 5 ⊢ (((1...(𝑁 − 1)) ∈ Fin ∧ 𝑅 ∈ V) → ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
18 | 12, 15, 17 | sylancr 581 | . . . 4 ⊢ (𝜑 → ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
19 | 18 | eleq2d 2892 | . . 3 ⊢ (𝜑 → (𝑆 ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))) |
20 | 11, 19 | mpbid 224 | . 2 ⊢ (𝜑 → 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
21 | smatcl.c | . 2 ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅)) | |
22 | 20, 21 | syl6eleqr 2917 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 × cxp 5344 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 Fincfn 8228 1c1 10260 − cmin 10592 ℕcn 11357 ...cfz 12626 Basecbs 16229 Mat cmat 20587 subMat1csmat 30400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-hom 16336 df-cco 16337 df-0g 16462 df-prds 16468 df-pws 16470 df-sra 19540 df-rgmod 19541 df-dsmm 20446 df-frlm 20461 df-mat 20588 df-smat 30401 |
This theorem is referenced by: submat1n 30412 submateq 30416 madjusmdetlem3 30436 mdetlap 30439 |
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