Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 12372 | . . . . 5 ⊢ (1...𝑀) ⊆ ℕ | |
8 | 7, 4 | sseldi 3601 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
9 | fzssnn 12385 | . . . 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 3604 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
13 | fzossnn 12516 | . . 3 ⊢ (1..^𝐿) ⊆ ℕ | |
14 | smattr.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) | |
15 | 13, 14 | sseldi 3601 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
16 | elfzle1 12344 | . . . . 5 ⊢ (𝐼 ∈ (𝐾...𝑀) → 𝐾 ≤ 𝐼) | |
17 | 11, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ≤ 𝐼) |
18 | 8 | nnred 11035 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
19 | 12 | nnred 11035 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
20 | 18, 19 | lenltd 10183 | . . . 4 ⊢ (𝜑 → (𝐾 ≤ 𝐼 ↔ ¬ 𝐼 < 𝐾)) |
21 | 17, 20 | mpbid 222 | . . 3 ⊢ (𝜑 → ¬ 𝐼 < 𝐾) |
22 | 21 | iffalsed 4097 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = (𝐼 + 1)) |
23 | elfzolt2 12479 | . . . 4 ⊢ (𝐽 ∈ (1..^𝐿) → 𝐽 < 𝐿) | |
24 | 14, 23 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 < 𝐿) |
25 | 24 | iftrued 4094 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝐽) |
26 | 1, 2, 3, 4, 5, 6, 12, 15, 22, 25 | smatlem 29863 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 ℕcn 11020 ...cfz 12326 ..^cfzo 12465 subMat1csmat 29859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-smat 29860 |
This theorem is referenced by: submateq 29875 |
Copyright terms: Public domain | W3C validator |