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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 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | smatcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
9 | 6, 7, 8 | matbas2i 22444 | . . . . 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 33757 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))) |
12 | fzfi 14010 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
13 | 6, 8 | matrcl 22432 | . . . . . . 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 2735 | . . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅) | |
17 | 16, 7 | matbas2 22443 | . . . . 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 587 | . . . 4 ⊢ (𝜑 → ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
19 | 18 | eleq2d 2825 | . . 3 ⊢ (𝜑 → (𝑆 ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))) |
20 | 11, 19 | mpbid 232 | . 2 ⊢ (𝜑 → 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
21 | smatcl.c | . 2 ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅)) | |
22 | 20, 21 | eleqtrrdi 2850 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 1c1 11154 − cmin 11490 ℕcn 12264 ...cfz 13544 Basecbs 17245 Mat cmat 22427 subMat1csmat 33754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-mat 22428 df-smat 33755 |
This theorem is referenced by: submat1n 33766 submateq 33770 madjusmdetlem3 33790 mdetlap 33793 |
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