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Mirrors > Home > MPE Home > Th. List > 2sqn0 | Structured version Visualization version GIF version |
Description: If the sum of two squares is prime, none of the original number is zero. (Contributed by Thierry Arnoux, 4-Feb-2020.) |
Ref | Expression |
---|---|
2sqcoprm.1 | ⊢ (𝜑 → 𝑃 ∈ ℙ) |
2sqcoprm.2 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
2sqcoprm.3 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
2sqcoprm.4 | ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃) |
Ref | Expression |
---|---|
2sqn0 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqcoprm.4 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃) | |
2 | 2sqcoprm.1 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
3 | 1, 2 | eqeltrd 2839 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
5 | sq0i 13899 | . . . . . 6 ⊢ (𝐴 = 0 → (𝐴↑2) = 0) | |
6 | 5 | oveq1d 7284 | . . . . 5 ⊢ (𝐴 = 0 → ((𝐴↑2) + (𝐵↑2)) = (0 + (𝐵↑2))) |
7 | 2sqcoprm.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
8 | 7 | zcnd 12416 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 8 | sqcld 13851 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
10 | 9 | addid2d 11165 | . . . . 5 ⊢ (𝜑 → (0 + (𝐵↑2)) = (𝐵↑2)) |
11 | 6, 10 | sylan9eqr 2800 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) = (𝐵↑2)) |
12 | sqnprm 16396 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → ¬ (𝐵↑2) ∈ ℙ) | |
13 | 7, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → ¬ (𝐵↑2) ∈ ℙ) |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ (𝐵↑2) ∈ ℙ) |
15 | 11, 14 | eqneltrd 2858 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
16 | 4, 15 | pm2.65da 814 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 0) |
17 | 16 | neqned 2950 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 (class class class)co 7269 0cc0 10860 + caddc 10863 2c2 12017 ℤcz 12308 ↑cexp 13771 ℙcprime 16365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-pre-sup 10938 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8487 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-sup 9190 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-n0 12223 df-z 12309 df-uz 12572 df-rp 12720 df-seq 13711 df-exp 13772 df-cj 14799 df-re 14800 df-im 14801 df-sqrt 14935 df-abs 14936 df-dvds 15953 df-prm 16366 |
This theorem is referenced by: 2sqcoprm 26572 2sqmod 26573 |
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