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Mirrors > Home > MPE Home > Th. List > zsqcl2 | Structured version Visualization version GIF version |
Description: The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
zsqcl2 | ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsqcl 13303 | . 2 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
2 | zre 11791 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | sqge0 13310 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ ℤ → 0 ≤ (𝐴↑2)) |
5 | elnn0z 11800 | . 2 ⊢ ((𝐴↑2) ∈ ℕ0 ↔ ((𝐴↑2) ∈ ℤ ∧ 0 ≤ (𝐴↑2))) | |
6 | 1, 4, 5 | sylanbrc 575 | 1 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 class class class wbr 4923 (class class class)co 6970 ℝcr 10328 0cc0 10329 ≤ cle 10469 2c2 11489 ℕ0cn0 11701 ℤcz 11787 ↑cexp 13238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-n0 11702 df-z 11788 df-uz 12053 df-seq 13179 df-exp 13239 |
This theorem is referenced by: znsqcld 13335 gzabssqcl 16127 4sqlem1 16134 4sqlem12 16142 4sqlem14 16144 4sqlem16 16146 lgsdir 25604 2sqlem7 25696 2sqlem8 25698 2sqblem 25703 |
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