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Mirrors > Home > MPE Home > Th. List > Mathboxes > smattr | Structured version Visualization version GIF version |
Description: Entries of a submatrix, top 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)))) |
smattr.i | ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀)) |
smattr.j | ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) |
Ref | Expression |
---|---|
smattr | ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽)) |
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 12932 | . . . . 5 ⊢ (1...𝑀) ⊆ ℕ | |
8 | 7, 4 | sseldi 3964 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
9 | fzssnn 12945 | . . . 4 ⊢ (𝐾 ∈ ℕ → (𝐾...𝑀) ⊆ ℕ) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾...𝑀) ⊆ ℕ) |
11 | smattr.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀)) | |
12 | 10, 11 | sseldd 3967 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
13 | fzossnn 13080 | . . 3 ⊢ (1..^𝐿) ⊆ ℕ | |
14 | smattr.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) | |
15 | 13, 14 | sseldi 3964 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
16 | elfzle1 12904 | . . . . 5 ⊢ (𝐼 ∈ (𝐾...𝑀) → 𝐾 ≤ 𝐼) | |
17 | 11, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ≤ 𝐼) |
18 | 8 | nnred 11647 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
19 | 12 | nnred 11647 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
20 | 18, 19 | lenltd 10780 | . . . 4 ⊢ (𝜑 → (𝐾 ≤ 𝐼 ↔ ¬ 𝐼 < 𝐾)) |
21 | 17, 20 | mpbid 234 | . . 3 ⊢ (𝜑 → ¬ 𝐼 < 𝐾) |
22 | 21 | iffalsed 4477 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = (𝐼 + 1)) |
23 | elfzolt2 13041 | . . . 4 ⊢ (𝐽 ∈ (1..^𝐿) → 𝐽 < 𝐿) | |
24 | 14, 23 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 < 𝐿) |
25 | 24 | iftrued 4474 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝐽) |
26 | 1, 2, 3, 4, 5, 6, 12, 15, 22, 25 | smatlem 31057 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 × cxp 5547 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 ℕcn 11632 ...cfz 12886 ..^cfzo 13027 subMat1csmat 31053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-smat 31054 |
This theorem is referenced by: submateq 31069 |
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