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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smattl | Structured version Visualization version GIF version |
Description: Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
Ref | Expression |
---|---|
smat.s | ⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) |
smat.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
smat.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
smat.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) |
smat.l | ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) |
smat.a | ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) |
smattl.i | ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) |
smattl.j | ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) |
Ref | Expression |
---|---|
smattl | ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smat.s | . 2 ⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) | |
2 | smat.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
3 | smat.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | smat.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) | |
5 | smat.l | . 2 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) | |
6 | smat.a | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) | |
7 | fzossnn 12819 | . . 3 ⊢ (1..^𝐾) ⊆ ℕ | |
8 | smattl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) | |
9 | 7, 8 | sseldi 3825 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
10 | fzossnn 12819 | . . 3 ⊢ (1..^𝐿) ⊆ ℕ | |
11 | smattl.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) | |
12 | 10, 11 | sseldi 3825 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
13 | elfzolt2 12781 | . . . 4 ⊢ (𝐼 ∈ (1..^𝐾) → 𝐼 < 𝐾) | |
14 | 8, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 < 𝐾) |
15 | 14 | iftrued 4316 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝐼) |
16 | elfzolt2 12781 | . . . 4 ⊢ (𝐽 ∈ (1..^𝐿) → 𝐽 < 𝐿) | |
17 | 11, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 < 𝐿) |
18 | 17 | iftrued 4316 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝐽) |
19 | 1, 2, 3, 4, 5, 6, 9, 12, 15, 18 | smatlem 30404 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 × cxp 5344 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 1c1 10260 + caddc 10262 < clt 10398 ℕcn 11357 ...cfz 12626 ..^cfzo 12767 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-uni 4661 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-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 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-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-smat 30401 |
This theorem is referenced by: submat1n 30412 submateq 30416 |
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