Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nnsqcld | Structured version Visualization version GIF version |
Description: The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
nnexpcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnsqcld | ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnexpcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnsqcl 13483 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℕ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7142 ℕcn 11624 2c2 11679 ↑cexp 13419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-n0 11885 df-z 11969 df-uz 12231 df-seq 13360 df-exp 13420 |
This theorem is referenced by: sqrt2irrlem 15586 sqgcd 15892 numdensq 16077 pythagtriplem4 16139 pythagtriplem19 16153 prmreclem1 16235 prmreclem3 16237 prmreclem6 16240 mul4sqlem 16272 4sqlem12 16275 4sqlem16 16279 lgamgulmlem3 25594 lgamgulmlem4 25595 lgamgulmlem6 25597 basellem8 25651 chpub 25782 2sqlem3 25982 2sqlem8 25988 2sqcoprm 25997 2sqmod 25998 dchrisum0fno1 26073 pellexlem2 39519 rmspecsqrtnq 39595 jm2.27a 39694 jm2.27c 39696 fmtnorec3 43795 |
Copyright terms: Public domain | W3C validator |