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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 2823 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | smatcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
9 | 6, 7, 8 | matbas2i 21033 | . . . . 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 31063 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))) |
12 | fzfi 13343 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
13 | 6, 8 | matrcl 21023 | . . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → ((1...𝑁) ∈ Fin ∧ 𝑅 ∈ V)) |
14 | 13 | simprd 498 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑅 ∈ V) |
15 | 5, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
16 | eqid 2823 | . . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅) | |
17 | 16, 7 | matbas2 21032 | . . . . 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 589 | . . . 4 ⊢ (𝜑 → ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
19 | 18 | eleq2d 2900 | . . 3 ⊢ (𝜑 → (𝑆 ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))) |
20 | 11, 19 | mpbid 234 | . 2 ⊢ (𝜑 → 𝑆 ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) |
21 | smatcl.c | . 2 ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅)) | |
22 | 20, 21 | eleqtrrdi 2926 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Fincfn 8511 1c1 10540 − cmin 10872 ℕcn 11640 ...cfz 12895 Basecbs 16485 Mat cmat 21018 subMat1csmat 31060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-prds 16723 df-pws 16725 df-sra 19946 df-rgmod 19947 df-dsmm 20878 df-frlm 20893 df-mat 21019 df-smat 31061 |
This theorem is referenced by: submat1n 31072 submateq 31076 madjusmdetlem3 31096 mdetlap 31099 |
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