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Mirrors > Home > MPE Home > Th. List > nnesq | Structured version Visualization version GIF version |
Description: A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
nnesq | ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ((𝑁↑2) / 2) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 12558 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | zesq 14168 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ((𝑁↑2) / 2) ∈ ℤ)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℤ ↔ ((𝑁↑2) / 2) ∈ ℤ)) |
4 | nnrp 12964 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
5 | 4 | rphalfcld 13007 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℝ+) |
6 | 5 | rpgt0d 12998 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 / 2)) |
7 | nnsqcl 14072 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℕ) | |
8 | 7 | nnrpd 12993 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℝ+) |
9 | 8 | rphalfcld 13007 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) / 2) ∈ ℝ+) |
10 | 9 | rpgt0d 12998 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < ((𝑁↑2) / 2)) |
11 | 6, 10 | 2thd 264 | . . 3 ⊢ (𝑁 ∈ ℕ → (0 < (𝑁 / 2) ↔ 0 < ((𝑁↑2) / 2))) |
12 | 3, 11 | anbi12d 631 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 / 2) ∈ ℤ ∧ 0 < (𝑁 / 2)) ↔ (((𝑁↑2) / 2) ∈ ℤ ∧ 0 < ((𝑁↑2) / 2)))) |
13 | elnnz 12547 | . 2 ⊢ ((𝑁 / 2) ∈ ℕ ↔ ((𝑁 / 2) ∈ ℤ ∧ 0 < (𝑁 / 2))) | |
14 | elnnz 12547 | . 2 ⊢ (((𝑁↑2) / 2) ∈ ℕ ↔ (((𝑁↑2) / 2) ∈ ℤ ∧ 0 < ((𝑁↑2) / 2))) | |
15 | 12, 13, 14 | 3bitr4g 313 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ((𝑁↑2) / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5138 (class class class)co 7390 0cc0 11089 < clt 11227 / cdiv 11850 ℕcn 12191 2c2 12246 ℤcz 12537 ↑cexp 14006 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-n0 12452 df-z 12538 df-uz 12802 df-rp 12954 df-seq 13946 df-exp 14007 |
This theorem is referenced by: sqrt2irrlem 16170 |
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