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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 12926 | . . . . 5 ⊢ (1...𝑀) ⊆ ℕ | |
8 | 7, 4 | sseldi 3962 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
9 | fzssnn 12939 | . . . 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 3965 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
13 | fzossnn 13074 | . . 3 ⊢ (1..^𝐿) ⊆ ℕ | |
14 | smattr.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) | |
15 | 13, 14 | sseldi 3962 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
16 | elfzle1 12898 | . . . . 5 ⊢ (𝐼 ∈ (𝐾...𝑀) → 𝐾 ≤ 𝐼) | |
17 | 11, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ≤ 𝐼) |
18 | 8 | nnred 11641 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
19 | 12 | nnred 11641 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
20 | 18, 19 | lenltd 10774 | . . . 4 ⊢ (𝜑 → (𝐾 ≤ 𝐼 ↔ ¬ 𝐼 < 𝐾)) |
21 | 17, 20 | mpbid 233 | . . 3 ⊢ (𝜑 → ¬ 𝐼 < 𝐾) |
22 | 21 | iffalsed 4474 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = (𝐼 + 1)) |
23 | elfzolt2 13035 | . . . 4 ⊢ (𝐽 ∈ (1..^𝐿) → 𝐽 < 𝐿) | |
24 | 14, 23 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 < 𝐿) |
25 | 24 | iftrued 4471 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝐽) |
26 | 1, 2, 3, 4, 5, 6, 12, 15, 22, 25 | smatlem 30961 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 ℕcn 11626 ...cfz 12880 ..^cfzo 13021 subMat1csmat 30957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-smat 30958 |
This theorem is referenced by: submateq 30973 |
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