![]() |
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 2859 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
4 | 3 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
5 | sq0i 13280 | . . . . . 6 ⊢ (𝐴 = 0 → (𝐴↑2) = 0) | |
6 | 5 | oveq1d 6939 | . . . . 5 ⊢ (𝐴 = 0 → ((𝐴↑2) + (𝐵↑2)) = (0 + (𝐵↑2))) |
7 | 2sqcoprm.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
8 | 7 | zcnd 11840 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 8 | sqcld 13330 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
10 | 9 | addid2d 10579 | . . . . 5 ⊢ (𝜑 → (0 + (𝐵↑2)) = (𝐵↑2)) |
11 | 6, 10 | sylan9eqr 2836 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ((𝐴↑2) + (𝐵↑2)) = (𝐵↑2)) |
12 | sqnprm 15829 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → ¬ (𝐵↑2) ∈ ℙ) | |
13 | 7, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → ¬ (𝐵↑2) ∈ ℙ) |
14 | 13 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ (𝐵↑2) ∈ ℙ) |
15 | 11, 14 | eqneltrd 2878 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → ¬ ((𝐴↑2) + (𝐵↑2)) ∈ ℙ) |
16 | 4, 15 | pm2.65da 807 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 0) |
17 | 16 | neqned 2976 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 (class class class)co 6924 0cc0 10274 + caddc 10277 2c2 11435 ℤcz 11733 ↑cexp 13183 ℙcprime 15800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-seq 13125 df-exp 13184 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-dvds 15397 df-prm 15801 |
This theorem is referenced by: 2sqcoprm 25622 2sqmod 25623 |
Copyright terms: Public domain | W3C validator |