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Mirrors > Home > ILE Home > Th. List > 2sqlem8a | GIF version |
Description: Lemma for 2sqlem8 14030. (Contributed by Mario Carneiro, 4-Jun-2016.) |
Ref | Expression |
---|---|
2sq.1 | ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) |
2sqlem7.2 | ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} |
2sqlem9.5 | ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) |
2sqlem9.7 | ⊢ (𝜑 → 𝑀 ∥ 𝑁) |
2sqlem8.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
2sqlem8.m | ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) |
2sqlem8.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
2sqlem8.2 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
2sqlem8.3 | ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) |
2sqlem8.4 | ⊢ (𝜑 → 𝑁 = ((𝐴↑2) + (𝐵↑2))) |
2sqlem8.c | ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) |
2sqlem8.d | ⊢ 𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) |
Ref | Expression |
---|---|
2sqlem8a | ⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqlem8.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 2sqlem8.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘2)) | |
3 | eluz2b3 9577 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1)) | |
4 | 2, 3 | sylib 122 | . . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1)) |
5 | 4 | simpld 112 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
6 | 2sqlem8.c | . . . 4 ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
7 | 1, 5, 6 | 4sqlem5 12347 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℤ ∧ ((𝐴 − 𝐶) / 𝑀) ∈ ℤ)) |
8 | 7 | simpld 112 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℤ) |
9 | 2sqlem8.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
10 | 2sqlem8.d | . . . 4 ⊢ 𝐷 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
11 | 9, 5, 10 | 4sqlem5 12347 | . . 3 ⊢ (𝜑 → (𝐷 ∈ ℤ ∧ ((𝐵 − 𝐷) / 𝑀) ∈ ℤ)) |
12 | 11 | simpld 112 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℤ) |
13 | 4 | simprd 114 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 1) |
14 | simpr 110 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝐶↑2) = 0) | |
15 | 1, 5, 6, 14 | 4sqlem9 12351 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝑀↑2) ∥ (𝐴↑2)) |
16 | 15 | ex 115 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶↑2) = 0 → (𝑀↑2) ∥ (𝐴↑2))) |
17 | eluzelz 9510 | . . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℤ) | |
18 | 2, 17 | syl 14 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
19 | dvdssq 11999 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) | |
20 | 18, 1, 19 | syl2anc 411 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) |
21 | 16, 20 | sylibrd 169 | . . . . . . 7 ⊢ (𝜑 → ((𝐶↑2) = 0 → 𝑀 ∥ 𝐴)) |
22 | simpr 110 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝐷↑2) = 0) | |
23 | 9, 5, 10, 22 | 4sqlem9 12351 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝑀↑2) ∥ (𝐵↑2)) |
24 | 23 | ex 115 | . . . . . . . 8 ⊢ (𝜑 → ((𝐷↑2) = 0 → (𝑀↑2) ∥ (𝐵↑2))) |
25 | dvdssq 11999 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2))) | |
26 | 18, 9, 25 | syl2anc 411 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2))) |
27 | 24, 26 | sylibrd 169 | . . . . . . 7 ⊢ (𝜑 → ((𝐷↑2) = 0 → 𝑀 ∥ 𝐵)) |
28 | 2sqlem8.3 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) | |
29 | 1ne0 8960 | . . . . . . . . . . . 12 ⊢ 1 ≠ 0 | |
30 | 29 | a1i 9 | . . . . . . . . . . 11 ⊢ (𝜑 → 1 ≠ 0) |
31 | 28, 30 | eqnetrd 2369 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≠ 0) |
32 | 31 | neneqd 2366 | . . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴 gcd 𝐵) = 0) |
33 | gcdeq0 11945 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | |
34 | 1, 9, 33 | syl2anc 411 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
35 | 32, 34 | mtbid 672 | . . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) |
36 | dvdslegcd 11932 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝑀 ∥ 𝐴 ∧ 𝑀 ∥ 𝐵) → 𝑀 ≤ (𝐴 gcd 𝐵))) | |
37 | 18, 1, 9, 35, 36 | syl31anc 1241 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 ∥ 𝐴 ∧ 𝑀 ∥ 𝐵) → 𝑀 ≤ (𝐴 gcd 𝐵))) |
38 | 21, 27, 37 | syl2and 295 | . . . . . 6 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 ≤ (𝐴 gcd 𝐵))) |
39 | 28 | breq2d 4010 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 ≤ 1)) |
40 | nnle1eq1 8916 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀 ≤ 1 ↔ 𝑀 = 1)) | |
41 | 5, 40 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ 1 ↔ 𝑀 = 1)) |
42 | 39, 41 | bitrd 188 | . . . . . 6 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 = 1)) |
43 | 38, 42 | sylibd 149 | . . . . 5 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 = 1)) |
44 | 43 | necon3ad 2387 | . . . 4 ⊢ (𝜑 → (𝑀 ≠ 1 → ¬ ((𝐶↑2) = 0 ∧ (𝐷↑2) = 0))) |
45 | 13, 44 | mpd 13 | . . 3 ⊢ (𝜑 → ¬ ((𝐶↑2) = 0 ∧ (𝐷↑2) = 0)) |
46 | 8 | zcnd 9349 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
47 | sqeq0 10553 | . . . . 5 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0)) | |
48 | 46, 47 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝐶↑2) = 0 ↔ 𝐶 = 0)) |
49 | 12 | zcnd 9349 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
50 | sqeq0 10553 | . . . . 5 ⊢ (𝐷 ∈ ℂ → ((𝐷↑2) = 0 ↔ 𝐷 = 0)) | |
51 | 49, 50 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝐷↑2) = 0 ↔ 𝐷 = 0)) |
52 | 48, 51 | anbi12d 473 | . . 3 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) ↔ (𝐶 = 0 ∧ 𝐷 = 0))) |
53 | 45, 52 | mtbid 672 | . 2 ⊢ (𝜑 → ¬ (𝐶 = 0 ∧ 𝐷 = 0)) |
54 | gcdn0cl 11930 | . 2 ⊢ (((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ ¬ (𝐶 = 0 ∧ 𝐷 = 0)) → (𝐶 gcd 𝐷) ∈ ℕ) | |
55 | 8, 12, 53, 54 | syl21anc 1237 | 1 ⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2146 {cab 2161 ≠ wne 2345 ∀wral 2453 ∃wrex 2454 class class class wbr 3998 ↦ cmpt 4059 ran crn 4621 ‘cfv 5208 (class class class)co 5865 ℂcc 7784 0cc0 7786 1c1 7787 + caddc 7789 ≤ cle 7967 − cmin 8102 / cdiv 8602 ℕcn 8892 2c2 8943 ℤcz 9226 ℤ≥cuz 9501 ...cfz 9979 mod cmo 10292 ↑cexp 10489 abscabs 10974 ∥ cdvds 11762 gcd cgcd 11910 ℤ[i]cgz 12334 |
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-stab 831 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-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 8603 df-inn 8893 df-2 8951 df-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-q 9593 df-rp 9625 df-fz 9980 df-fzo 10113 df-fl 10240 df-mod 10293 df-seqfrec 10416 df-exp 10490 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 df-dvds 11763 df-gcd 11911 |
This theorem is referenced by: 2sqlem8 14030 |
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