![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2825 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
5 | sq0i 14155 | . . . . . 6 ⊢ (𝐴 = 0 → (𝐴↑2) = 0) | |
6 | 5 | oveq1d 7417 | . . . . 5 ⊢ (𝐴 = 0 → ((𝐴↑2) + (𝐵↑2)) = (0 + (𝐵↑2))) |
7 | 2sqcoprm.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
8 | 7 | zcnd 12665 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 8 | sqcld 14107 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
10 | 9 | addlidd 11413 | . . . . 5 ⊢ (𝜑 → (0 + (𝐵↑2)) = (𝐵↑2)) |
11 | 6, 10 | sylan9eqr 2786 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) = (𝐵↑2)) |
12 | sqnprm 16638 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → ¬ (𝐵↑2) ∈ ℙ) | |
13 | 7, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → ¬ (𝐵↑2) ∈ ℙ) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ (𝐵↑2) ∈ ℙ) |
15 | 11, 14 | eqneltrd 2845 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
16 | 4, 15 | pm2.65da 814 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 0) |
17 | 16 | neqned 2939 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 (class class class)co 7402 0cc0 11107 + caddc 11110 2c2 12265 ℤcz 12556 ↑cexp 14025 ℙcprime 16607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 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 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-dvds 16197 df-prm 16608 |
This theorem is referenced by: 2sqcoprm 27287 2sqmod 27288 |
Copyright terms: Public domain | W3C validator |