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Mathbox for Thierry Arnoux |
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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 13684 | . . 3 ⊢ (1..^𝐾) ⊆ ℕ | |
8 | smatbl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) | |
9 | 7, 8 | sselid 3975 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
10 | fz1ssnn 13535 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
11 | 10, 5 | sselid 3975 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ) |
12 | fzssnn 13548 | . . . 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 3978 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
16 | elfzolt2 13644 | . . . 4 ⊢ (𝐼 ∈ (1..^𝐾) → 𝐼 < 𝐾) | |
17 | 8, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 < 𝐾) |
18 | 17 | iftrued 4531 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝐼) |
19 | elfzle1 13507 | . . . . 5 ⊢ (𝐽 ∈ (𝐿...𝑁) → 𝐿 ≤ 𝐽) | |
20 | 14, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ≤ 𝐽) |
21 | 11 | nnred 12228 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
22 | 15 | nnred 12228 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
23 | 21, 22 | lenltd 11361 | . . . 4 ⊢ (𝜑 → (𝐿 ≤ 𝐽 ↔ ¬ 𝐽 < 𝐿)) |
24 | 20, 23 | mpbid 231 | . . 3 ⊢ (𝜑 → ¬ 𝐽 < 𝐿) |
25 | 24 | iffalsed 4534 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = (𝐽 + 1)) |
26 | 1, 2, 3, 4, 5, 6, 9, 15, 18, 25 | smatlem 33307 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 class class class wbr 5141 × cxp 5667 ‘cfv 6536 (class class class)co 7404 ↑m cmap 8819 1c1 11110 + caddc 11112 < clt 11249 ≤ cle 11250 ℕcn 12213 ...cfz 13487 ..^cfzo 13630 subMat1csmat 33303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-smat 33304 |
This theorem is referenced by: submateq 33319 |
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