Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smatbr | Structured version Visualization version GIF version |
Description: Entries of a submatrix, bottom right. (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)))) |
smatbr.i | ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀)) |
smatbr.j | ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁)) |
Ref | Expression |
---|---|
smatbr | ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(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 | fz1ssnn 13032 | . . . . 5 ⊢ (1...𝑀) ⊆ ℕ | |
8 | 7, 4 | sseldi 3876 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
9 | fzssnn 13045 | . . . 4 ⊢ (𝐾 ∈ ℕ → (𝐾...𝑀) ⊆ ℕ) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾...𝑀) ⊆ ℕ) |
11 | smatbr.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀)) | |
12 | 10, 11 | sseldd 3879 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
13 | fz1ssnn 13032 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
14 | 13, 5 | sseldi 3876 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ) |
15 | fzssnn 13045 | . . . 4 ⊢ (𝐿 ∈ ℕ → (𝐿...𝑁) ⊆ ℕ) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → (𝐿...𝑁) ⊆ ℕ) |
17 | smatbr.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁)) | |
18 | 16, 17 | sseldd 3879 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
19 | elfzle1 13004 | . . . . 5 ⊢ (𝐼 ∈ (𝐾...𝑀) → 𝐾 ≤ 𝐼) | |
20 | 11, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ≤ 𝐼) |
21 | 8 | nnred 11734 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
22 | 12 | nnred 11734 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
23 | 21, 22 | lenltd 10867 | . . . 4 ⊢ (𝜑 → (𝐾 ≤ 𝐼 ↔ ¬ 𝐼 < 𝐾)) |
24 | 20, 23 | mpbid 235 | . . 3 ⊢ (𝜑 → ¬ 𝐼 < 𝐾) |
25 | 24 | iffalsed 4426 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = (𝐼 + 1)) |
26 | elfzle1 13004 | . . . . 5 ⊢ (𝐽 ∈ (𝐿...𝑁) → 𝐿 ≤ 𝐽) | |
27 | 17, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ≤ 𝐽) |
28 | 14 | nnred 11734 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
29 | 18 | nnred 11734 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
30 | 28, 29 | lenltd 10867 | . . . 4 ⊢ (𝜑 → (𝐿 ≤ 𝐽 ↔ ¬ 𝐽 < 𝐿)) |
31 | 27, 30 | mpbid 235 | . . 3 ⊢ (𝜑 → ¬ 𝐽 < 𝐿) |
32 | 31 | iffalsed 4426 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = (𝐽 + 1)) |
33 | 1, 2, 3, 4, 5, 6, 12, 18, 25, 32 | smatlem 31322 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3844 class class class wbr 5031 × cxp 5524 ‘cfv 6340 (class class class)co 7173 ↑m cmap 8440 1c1 10619 + caddc 10621 < clt 10756 ≤ cle 10757 ℕcn 11719 ...cfz 12984 subMat1csmat 31318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-z 12066 df-uz 12328 df-fz 12985 df-smat 31319 |
This theorem is referenced by: submateq 31334 |
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