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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 13735 | . . 3 ⊢ (1..^𝐾) ⊆ ℕ | |
8 | smattl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) | |
9 | 7, 8 | sselid 3977 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℕ) |
10 | fzossnn 13735 | . . 3 ⊢ (1..^𝐿) ⊆ ℕ | |
11 | smattl.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) | |
12 | 10, 11 | sselid 3977 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ) |
13 | elfzolt2 13695 | . . . 4 ⊢ (𝐼 ∈ (1..^𝐾) → 𝐼 < 𝐾) | |
14 | 8, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 < 𝐾) |
15 | 14 | iftrued 4541 | . 2 ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝐼) |
16 | elfzolt2 13695 | . . . 4 ⊢ (𝐽 ∈ (1..^𝐿) → 𝐽 < 𝐿) | |
17 | 11, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 < 𝐿) |
18 | 17 | iftrued 4541 | . 2 ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝐽) |
19 | 1, 2, 3, 4, 5, 6, 9, 12, 15, 18 | smatlem 33612 | 1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 × cxp 5680 ‘cfv 6554 (class class class)co 7424 ↑m cmap 8855 1c1 11159 + caddc 11161 < clt 11298 ℕcn 12264 ...cfz 13538 ..^cfzo 13681 subMat1csmat 33608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-smat 33609 |
This theorem is referenced by: submat1n 33620 submateq 33624 |
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