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Mirrors > Home > ILE Home > Th. List > 2sqlem5 | GIF version |
Description: Lemma for 2sq . If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Ref | Expression |
---|---|
2sq.1 | ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) |
2sqlem5.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
2sqlem5.2 | ⊢ (𝜑 → 𝑃 ∈ ℙ) |
2sqlem5.3 | ⊢ (𝜑 → (𝑁 · 𝑃) ∈ 𝑆) |
2sqlem5.4 | ⊢ (𝜑 → 𝑃 ∈ 𝑆) |
Ref | Expression |
---|---|
2sqlem5 | ⊢ (𝜑 → 𝑁 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqlem5.4 | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑆) | |
2 | 2sq.1 | . . . 4 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) | |
3 | 2 | 2sqlem2 14020 | . . 3 ⊢ (𝑃 ∈ 𝑆 ↔ ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2))) |
4 | 1, 3 | sylib 122 | . 2 ⊢ (𝜑 → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2))) |
5 | 2sqlem5.3 | . . 3 ⊢ (𝜑 → (𝑁 · 𝑃) ∈ 𝑆) | |
6 | 2 | 2sqlem2 14020 | . . 3 ⊢ ((𝑁 · 𝑃) ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) |
7 | 5, 6 | sylib 122 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) |
8 | reeanv 2644 | . . 3 ⊢ (∃𝑝 ∈ ℤ ∃𝑥 ∈ ℤ (∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) ↔ (∃𝑝 ∈ ℤ ∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2)))) | |
9 | reeanv 2644 | . . . . 5 ⊢ (∃𝑞 ∈ ℤ ∃𝑦 ∈ ℤ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) ↔ (∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2)))) | |
10 | 2sqlem5.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
11 | 10 | ad2antrr 488 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑁 ∈ ℕ) |
12 | 2sqlem5.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
13 | 12 | ad2antrr 488 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑃 ∈ ℙ) |
14 | simplrr 536 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑥 ∈ ℤ) | |
15 | simprlr 538 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑦 ∈ ℤ) | |
16 | simplrl 535 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑝 ∈ ℤ) | |
17 | simprll 537 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑞 ∈ ℤ) | |
18 | simprrr 540 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) | |
19 | simprrl 539 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑃 = ((𝑝↑2) + (𝑞↑2))) | |
20 | 2, 11, 13, 14, 15, 16, 17, 18, 19 | 2sqlem4 14023 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ ((𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))))) → 𝑁 ∈ 𝑆) |
21 | 20 | expr 375 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) ∧ (𝑞 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) → 𝑁 ∈ 𝑆)) |
22 | 21 | rexlimdvva 2600 | . . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (∃𝑞 ∈ ℤ ∃𝑦 ∈ ℤ (𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) → 𝑁 ∈ 𝑆)) |
23 | 9, 22 | syl5bir 153 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) → 𝑁 ∈ 𝑆)) |
24 | 23 | rexlimdvva 2600 | . . 3 ⊢ (𝜑 → (∃𝑝 ∈ ℤ ∃𝑥 ∈ ℤ (∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) → 𝑁 ∈ 𝑆)) |
25 | 8, 24 | syl5bir 153 | . 2 ⊢ (𝜑 → ((∃𝑝 ∈ ℤ ∃𝑞 ∈ ℤ 𝑃 = ((𝑝↑2) + (𝑞↑2)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝑁 · 𝑃) = ((𝑥↑2) + (𝑦↑2))) → 𝑁 ∈ 𝑆)) |
26 | 4, 7, 25 | mp2and 433 | 1 ⊢ (𝜑 → 𝑁 ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2146 ∃wrex 2454 ↦ cmpt 4059 ran crn 4621 ‘cfv 5208 (class class class)co 5865 + caddc 7789 · cmul 7791 ℕcn 8890 2c2 8941 ℤcz 9224 ↑cexp 10487 abscabs 10972 ℙcprime 12072 ℤ[i]cgz 12332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-1o 6407 df-2o 6408 df-er 6525 df-en 6731 df-sup 6973 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-q 9591 df-rp 9623 df-fz 9978 df-fzo 10111 df-fl 10238 df-mod 10291 df-seqfrec 10414 df-exp 10488 df-cj 10817 df-re 10818 df-im 10819 df-rsqrt 10973 df-abs 10974 df-dvds 11761 df-gcd 11909 df-prm 12073 df-gz 12333 |
This theorem is referenced by: 2sqlem6 14025 |
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