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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smattl | Structured version Visualization version GIF version |
Description: Entries of a submatrix, top 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)))) |
smattl.i | ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) |
smattl.j | ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) |
Ref | Expression |
---|---|
smattl | ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽)) |
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 13081 | . . 3 ⊢ (1..^𝐾) ⊆ ℕ | |
8 | smattl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) | |
9 | 7, 8 | sseldi 3913 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
10 | fzossnn 13081 | . . 3 ⊢ (1..^𝐿) ⊆ ℕ | |
11 | smattl.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) | |
12 | 10, 11 | sseldi 3913 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
13 | elfzolt2 13042 | . . . 4 ⊢ (𝐼 ∈ (1..^𝐾) → 𝐼 < 𝐾) | |
14 | 8, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 < 𝐾) |
15 | 14 | iftrued 4433 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝐼) |
16 | elfzolt2 13042 | . . . 4 ⊢ (𝐽 ∈ (1..^𝐿) → 𝐽 < 𝐿) | |
17 | 11, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 < 𝐿) |
18 | 17 | iftrued 4433 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝐽) |
19 | 1, 2, 3, 4, 5, 6, 9, 12, 15, 18 | smatlem 31150 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 1c1 10527 + caddc 10529 < clt 10664 ℕcn 11625 ...cfz 12885 ..^cfzo 13028 subMat1csmat 31146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-smat 31147 |
This theorem is referenced by: submat1n 31158 submateq 31162 |
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