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Mirrors > Home > MPE Home > Th. List > 2sqlem8a | Structured version Visualization version GIF version |
Description: Lemma for 2sqlem8 26777. (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 12848 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1)) | |
4 | 2, 3 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1)) |
5 | 4 | simpld 496 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
6 | 2sqlem8.c | . . . 4 ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
7 | 1, 5, 6 | 4sqlem5 16815 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℤ ∧ ((𝐴 − 𝐶) / 𝑀) ∈ ℤ)) |
8 | 7 | simpld 496 | . 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 16815 | . . 3 ⊢ (𝜑 → (𝐷 ∈ ℤ ∧ ((𝐵 − 𝐷) / 𝑀) ∈ ℤ)) |
12 | 11 | simpld 496 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℤ) |
13 | 4 | simprd 497 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 1) |
14 | simpr 486 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝐶↑2) = 0) | |
15 | 1, 5, 6, 14 | 4sqlem9 16819 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝑀↑2) ∥ (𝐴↑2)) |
16 | 15 | ex 414 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶↑2) = 0 → (𝑀↑2) ∥ (𝐴↑2))) |
17 | eluzelz 12774 | . . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℤ) | |
18 | 2, 17 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
19 | dvdssq 16444 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) | |
20 | 18, 1, 19 | syl2anc 585 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) |
21 | 16, 20 | sylibrd 259 | . . . . . . 7 ⊢ (𝜑 → ((𝐶↑2) = 0 → 𝑀 ∥ 𝐴)) |
22 | simpr 486 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝐷↑2) = 0) | |
23 | 9, 5, 10, 22 | 4sqlem9 16819 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝑀↑2) ∥ (𝐵↑2)) |
24 | 23 | ex 414 | . . . . . . . 8 ⊢ (𝜑 → ((𝐷↑2) = 0 → (𝑀↑2) ∥ (𝐵↑2))) |
25 | dvdssq 16444 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2))) | |
26 | 18, 9, 25 | syl2anc 585 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2))) |
27 | 24, 26 | sylibrd 259 | . . . . . . 7 ⊢ (𝜑 → ((𝐷↑2) = 0 → 𝑀 ∥ 𝐵)) |
28 | 2sqlem8.3 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) | |
29 | ax-1ne0 11121 | . . . . . . . . . . . 12 ⊢ 1 ≠ 0 | |
30 | 29 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝜑 → 1 ≠ 0) |
31 | 28, 30 | eqnetrd 3012 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≠ 0) |
32 | 31 | neneqd 2949 | . . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴 gcd 𝐵) = 0) |
33 | gcdeq0 16398 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | |
34 | 1, 9, 33 | syl2anc 585 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
35 | 32, 34 | mtbid 324 | . . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) |
36 | dvdslegcd 16385 | . . . . . . . 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 1374 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 ∥ 𝐴 ∧ 𝑀 ∥ 𝐵) → 𝑀 ≤ (𝐴 gcd 𝐵))) |
38 | 21, 27, 37 | syl2and 609 | . . . . . 6 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 ≤ (𝐴 gcd 𝐵))) |
39 | 28 | breq2d 5118 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 ≤ 1)) |
40 | nnle1eq1 12184 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀 ≤ 1 ↔ 𝑀 = 1)) | |
41 | 5, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ 1 ↔ 𝑀 = 1)) |
42 | 39, 41 | bitrd 279 | . . . . . 6 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 = 1)) |
43 | 38, 42 | sylibd 238 | . . . . 5 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 = 1)) |
44 | 43 | necon3ad 2957 | . . . 4 ⊢ (𝜑 → (𝑀 ≠ 1 → ¬ ((𝐶↑2) = 0 ∧ (𝐷↑2) = 0))) |
45 | 13, 44 | mpd 15 | . . 3 ⊢ (𝜑 → ¬ ((𝐶↑2) = 0 ∧ (𝐷↑2) = 0)) |
46 | 8 | zcnd 12609 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
47 | sqeq0 14026 | . . . . 5 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0)) | |
48 | 46, 47 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐶↑2) = 0 ↔ 𝐶 = 0)) |
49 | 12 | zcnd 12609 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
50 | sqeq0 14026 | . . . . 5 ⊢ (𝐷 ∈ ℂ → ((𝐷↑2) = 0 ↔ 𝐷 = 0)) | |
51 | 49, 50 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐷↑2) = 0 ↔ 𝐷 = 0)) |
52 | 48, 51 | anbi12d 632 | . . 3 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) ↔ (𝐶 = 0 ∧ 𝐷 = 0))) |
53 | 45, 52 | mtbid 324 | . 2 ⊢ (𝜑 → ¬ (𝐶 = 0 ∧ 𝐷 = 0)) |
54 | gcdn0cl 16383 | . 2 ⊢ (((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ ¬ (𝐶 = 0 ∧ 𝐷 = 0)) → (𝐶 gcd 𝐷) ∈ ℕ) | |
55 | 8, 12, 53, 54 | syl21anc 837 | 1 ⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2714 ≠ wne 2944 ∀wral 3065 ∃wrex 3074 class class class wbr 5106 ↦ cmpt 5189 ran crn 5635 ‘cfv 6497 (class class class)co 7358 ℂcc 11050 0cc0 11052 1c1 11053 + caddc 11055 ≤ cle 11191 − cmin 11386 / cdiv 11813 ℕcn 12154 2c2 12209 ℤcz 12500 ℤ≥cuz 12764 ...cfz 13425 mod cmo 13775 ↑cexp 13968 abscabs 15120 ∥ cdvds 16137 gcd cgcd 16375 ℤ[i]cgz 16802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-fl 13698 df-mod 13776 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-dvds 16138 df-gcd 16376 |
This theorem is referenced by: 2sqlem8 26777 |
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