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 13080 | . . 3 ⊢ (1..^𝐾) ⊆ ℕ | |
8 | smattl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) | |
9 | 7, 8 | sseldi 3965 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
10 | fzossnn 13080 | . . 3 ⊢ (1..^𝐿) ⊆ ℕ | |
11 | smattl.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) | |
12 | 10, 11 | sseldi 3965 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
13 | elfzolt2 13041 | . . . 4 ⊢ (𝐼 ∈ (1..^𝐾) → 𝐼 < 𝐾) | |
14 | 8, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 < 𝐾) |
15 | 14 | iftrued 4475 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝐼) |
16 | elfzolt2 13041 | . . . 4 ⊢ (𝐽 ∈ (1..^𝐿) → 𝐽 < 𝐿) | |
17 | 11, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 < 𝐿) |
18 | 17 | iftrued 4475 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝐽) |
19 | 1, 2, 3, 4, 5, 6, 9, 12, 15, 18 | smatlem 31057 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 × cxp 5548 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 1c1 10532 + caddc 10534 < clt 10669 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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 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: submat1n 31065 submateq 31069 |
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