![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sqnprm | Structured version Visualization version GIF version |
Description: A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Ref | Expression |
---|---|
sqnprm | ⊢ (𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 12615 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → 𝐴 ∈ ℝ) |
3 | absresq 15338 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → ((abs‘𝐴)↑2) = (𝐴↑2)) |
5 | 2 | recnd 11287 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → 𝐴 ∈ ℂ) |
6 | 5 | abscld 15472 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (abs‘𝐴) ∈ ℝ) |
7 | 6 | recnd 11287 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (abs‘𝐴) ∈ ℂ) |
8 | 7 | sqvald 14180 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → ((abs‘𝐴)↑2) = ((abs‘𝐴) · (abs‘𝐴))) |
9 | 4, 8 | eqtr3d 2777 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (𝐴↑2) = ((abs‘𝐴) · (abs‘𝐴))) |
10 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (𝐴↑2) ∈ ℙ) | |
11 | 9, 10 | eqeltrrd 2840 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → ((abs‘𝐴) · (abs‘𝐴)) ∈ ℙ) |
12 | nn0abscl 15348 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0) | |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (abs‘𝐴) ∈ ℕ0) |
14 | 13 | nn0zd 12637 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (abs‘𝐴) ∈ ℤ) |
15 | sq1 14231 | . . . . . 6 ⊢ (1↑2) = 1 | |
16 | prmuz2 16730 | . . . . . . . . 9 ⊢ ((𝐴↑2) ∈ ℙ → (𝐴↑2) ∈ (ℤ≥‘2)) | |
17 | 16 | adantl 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (𝐴↑2) ∈ (ℤ≥‘2)) |
18 | eluz2gt1 12960 | . . . . . . . 8 ⊢ ((𝐴↑2) ∈ (ℤ≥‘2) → 1 < (𝐴↑2)) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → 1 < (𝐴↑2)) |
20 | 19, 4 | breqtrrd 5176 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → 1 < ((abs‘𝐴)↑2)) |
21 | 15, 20 | eqbrtrid 5183 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (1↑2) < ((abs‘𝐴)↑2)) |
22 | 5 | absge0d 15480 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → 0 ≤ (abs‘𝐴)) |
23 | 1re 11259 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
24 | 0le1 11784 | . . . . . . 7 ⊢ 0 ≤ 1 | |
25 | lt2sq 14170 | . . . . . . 7 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 < (abs‘𝐴) ↔ (1↑2) < ((abs‘𝐴)↑2))) | |
26 | 23, 24, 25 | mpanl12 702 | . . . . . 6 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (1 < (abs‘𝐴) ↔ (1↑2) < ((abs‘𝐴)↑2))) |
27 | 6, 22, 26 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (1 < (abs‘𝐴) ↔ (1↑2) < ((abs‘𝐴)↑2))) |
28 | 21, 27 | mpbird 257 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → 1 < (abs‘𝐴)) |
29 | eluz2b1 12959 | . . . 4 ⊢ ((abs‘𝐴) ∈ (ℤ≥‘2) ↔ ((abs‘𝐴) ∈ ℤ ∧ 1 < (abs‘𝐴))) | |
30 | 14, 28, 29 | sylanbrc 583 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → (abs‘𝐴) ∈ (ℤ≥‘2)) |
31 | nprm 16722 | . . 3 ⊢ (((abs‘𝐴) ∈ (ℤ≥‘2) ∧ (abs‘𝐴) ∈ (ℤ≥‘2)) → ¬ ((abs‘𝐴) · (abs‘𝐴)) ∈ ℙ) | |
32 | 30, 30, 31 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴↑2) ∈ ℙ) → ¬ ((abs‘𝐴) · (abs‘𝐴)) ∈ ℙ) |
33 | 11, 32 | pm2.65da 817 | 1 ⊢ (𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ℤ≥cuz 12876 ↑cexp 14099 abscabs 15270 ℙcprime 16705 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-prm 16706 |
This theorem is referenced by: 2sqblem 27490 2sqn0 27493 2sqcoprm 27494 2sqnn 27498 |
Copyright terms: Public domain | W3C validator |