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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 2911 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
5 | sq0i 13548 | . . . . . 6 ⊢ (𝐴 = 0 → (𝐴↑2) = 0) | |
6 | 5 | oveq1d 7163 | . . . . 5 ⊢ (𝐴 = 0 → ((𝐴↑2) + (𝐵↑2)) = (0 + (𝐵↑2))) |
7 | 2sqcoprm.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
8 | 7 | zcnd 12080 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 8 | sqcld 13500 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
10 | 9 | addid2d 10833 | . . . . 5 ⊢ (𝜑 → (0 + (𝐵↑2)) = (𝐵↑2)) |
11 | 6, 10 | sylan9eqr 2876 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) = (𝐵↑2)) |
12 | sqnprm 16038 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → ¬ (𝐵↑2) ∈ ℙ) | |
13 | 7, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → ¬ (𝐵↑2) ∈ ℙ) |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ (𝐵↑2) ∈ ℙ) |
15 | 11, 14 | eqneltrd 2930 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
16 | 4, 15 | pm2.65da 815 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 0) |
17 | 16 | neqned 3021 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 (class class class)co 7148 0cc0 10529 + caddc 10532 2c2 11684 ℤcz 11973 ↑cexp 13421 ℙcprime 16007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16008 |
This theorem is referenced by: 2sqcoprm 26003 2sqmod 26004 |
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