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Mirrors > Home > MPE Home > Th. List > Mathboxes > smatbl | Structured version Visualization version GIF version |
Description: Entries of a submatrix, bottom 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)))) |
smatbl.i | ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) |
smatbl.j | ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁)) |
Ref | Expression |
---|---|
smatbl | ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1))) |
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 12836 | . . 3 ⊢ (1..^𝐾) ⊆ ℕ | |
8 | smatbl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) | |
9 | 7, 8 | sseldi 3818 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
10 | fz1ssnn 12689 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
11 | 10, 5 | sseldi 3818 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ) |
12 | fzssnn 12702 | . . . 4 ⊢ (𝐿 ∈ ℕ → (𝐿...𝑁) ⊆ ℕ) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐿...𝑁) ⊆ ℕ) |
14 | smatbl.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁)) | |
15 | 13, 14 | sseldd 3821 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
16 | elfzolt2 12798 | . . . 4 ⊢ (𝐼 ∈ (1..^𝐾) → 𝐼 < 𝐾) | |
17 | 8, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 < 𝐾) |
18 | 17 | iftrued 4314 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝐼) |
19 | elfzle1 12661 | . . . . 5 ⊢ (𝐽 ∈ (𝐿...𝑁) → 𝐿 ≤ 𝐽) | |
20 | 14, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ≤ 𝐽) |
21 | 11 | nnred 11391 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
22 | 15 | nnred 11391 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
23 | 21, 22 | lenltd 10522 | . . . 4 ⊢ (𝜑 → (𝐿 ≤ 𝐽 ↔ ¬ 𝐽 < 𝐿)) |
24 | 20, 23 | mpbid 224 | . . 3 ⊢ (𝜑 → ¬ 𝐽 < 𝐿) |
25 | 24 | iffalsed 4317 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = (𝐽 + 1)) |
26 | 1, 2, 3, 4, 5, 6, 9, 15, 18, 25 | smatlem 30461 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 class class class wbr 4886 × cxp 5353 ‘cfv 6135 (class class class)co 6922 ↑𝑚 cmap 8140 1c1 10273 + caddc 10275 < clt 10411 ≤ cle 10412 ℕcn 11374 ...cfz 12643 ..^cfzo 12784 subMat1csmat 30457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-smat 30458 |
This theorem is referenced by: submateq 30473 |
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