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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 2728 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | smatcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
9 | 6, 7, 8 | matbas2i 22337 | . . . . 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 33397 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))) |
12 | fzfi 13970 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
13 | 6, 8 | matrcl 22325 | . . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → ((1...𝑁) ∈ Fin ∧ 𝑅 ∈ V)) |
14 | 13 | simprd 495 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑅 ∈ V) |
15 | 5, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
16 | eqid 2728 | . . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅) | |
17 | 16, 7 | matbas2 22336 | . . . . 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 586 | . . . 4 ⊢ (𝜑 → ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
19 | 18 | eleq2d 2815 | . . 3 ⊢ (𝜑 → (𝑆 ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))) |
20 | 11, 19 | mpbid 231 | . 2 ⊢ (𝜑 → 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
21 | smatcl.c | . 2 ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅)) | |
22 | 20, 21 | eleqtrrdi 2840 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 × cxp 5676 ‘cfv 6548 (class class class)co 7420 ↑m cmap 8845 Fincfn 8964 1c1 11140 − cmin 11475 ℕcn 12243 ...cfz 13517 Basecbs 17180 Mat cmat 22320 subMat1csmat 33394 |
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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 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-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-iun 4998 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-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-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 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-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 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-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-prds 17429 df-pws 17431 df-sra 21058 df-rgmod 21059 df-dsmm 21666 df-frlm 21681 df-mat 22321 df-smat 33395 |
This theorem is referenced by: submat1n 33406 submateq 33410 madjusmdetlem3 33430 mdetlap 33433 |
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