2sqlem8a - Metamath Proof Explorer

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Theorem 2sqlem8a 27752

Description: Lemma for 2sqlem8 27753. (Contributed by Mario Carneiro, 4-Jun-2016.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
2sqlem8a	⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ)

Distinct variable groups: 𝑎,𝑏,𝑤,𝑥,𝑦,𝑧 𝐴,𝑎,𝑥,𝑦,𝑧 𝑥,𝐶 𝜑,𝑥,𝑦 𝐵,𝑎,𝑏,𝑥,𝑦 𝑀,𝑎,𝑏,𝑥,𝑦,𝑧 𝑆,𝑎,𝑏,𝑥,𝑦,𝑧 𝑥,𝐷 𝑥,𝑁,𝑦,𝑧 𝑌,𝑎,𝑏,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑧, 𝑤, 𝑎, 𝑏) 𝐴(𝑤, 𝑏) 𝐵(𝑧, 𝑤) 𝐶(𝑦, 𝑧, 𝑤, 𝑎, 𝑏) 𝐷(𝑦, 𝑧, 𝑤, 𝑎, 𝑏) 𝑆(𝑤) 𝑀(𝑤) 𝑁(𝑤, 𝑎, 𝑏) 𝑌(𝑧, 𝑤)

**Proof of Theorem 2sqlem8a**
Step	Hyp	Ref	Expression
1		2sqlem8.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ)
2		2sqlem8.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘2))
3		eluz2b3 13049	. . . . . 6 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
4	2, 3	sylib 221	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
5	4	simpld 500	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6		2sqlem8.c	. . . 4 ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
7	1, 5, 6	4sqlem5 17120	. . 3 ⊢ (𝜑 → (𝐶 ∈ ℤ ∧ ((𝐴 − 𝐶) / 𝑀) ∈ ℤ))
8	7	simpld 500	. 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 17120	. . 3 ⊢ (𝜑 → (𝐷 ∈ ℤ ∧ ((𝐵 − 𝐷) / 𝑀) ∈ ℤ))
12	11	simpld 500	. 2 ⊢ (𝜑 → 𝐷 ∈ ℤ)
13	4	simprd 501	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 1)
14		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝐶↑2) = 0)
15	1, 5, 6, 14	4sqlem9 17124	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝑀↑2) ∥ (𝐴↑2))
16	15	ex 418	. . . . . . . 8 ⊢ (𝜑 → ((𝐶↑2) = 0 → (𝑀↑2) ∥ (𝐴↑2)))
17		eluzelz 12975	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
18	2, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19		dvdssq 16742	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2)))
20	18, 1, 19	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2)))
21	16, 20	sylibrd 262	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑2) = 0 → 𝑀 ∥ 𝐴))
22		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝐷↑2) = 0)
23	9, 5, 10, 22	4sqlem9 17124	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝑀↑2) ∥ (𝐵↑2))
24	23	ex 418	. . . . . . . 8 ⊢ (𝜑 → ((𝐷↑2) = 0 → (𝑀↑2) ∥ (𝐵↑2)))
25		dvdssq 16742	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2)))
26	18, 9, 25	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2)))
27	24, 26	sylibrd 262	. . . . . . 7 ⊢ (𝜑 → ((𝐷↑2) = 0 → 𝑀 ∥ 𝐵))
28		2sqlem8.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)
29		ax-1ne0 11269	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≠ 0)
31	28, 30	eqnetrd 3023	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≠ 0)
32	31	neneqd 2961	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴 gcd 𝐵) = 0)
33		gcdeq0 16689	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
34	1, 9, 33	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
35	32, 34	mtbid 327	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
36		dvdslegcd 16674	. . . . . . . 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 1400	. . . . . . 7 ⊢ (𝜑 → ((𝑀 ∥ 𝐴 ∧ 𝑀 ∥ 𝐵) → 𝑀 ≤ (𝐴 gcd 𝐵)))
38	21, 27, 37	syl2and 620	. . . . . 6 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 ≤ (𝐴 gcd 𝐵)))
39	28	breq2d 5115	. . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 ≤ 1))
40		nnle1eq1 12368	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀 ≤ 1 ↔ 𝑀 = 1))
41	5, 40	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ 1 ↔ 𝑀 = 1))
42	39, 41	bitrd 282	. . . . . 6 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 = 1))
43	38, 42	sylibd 242	. . . . 5 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 = 1))
44	43	necon3ad 2969	. . . 4 ⊢ (𝜑 → (𝑀 ≠ 1 → ¬ ((𝐶↑2) = 0 ∧ (𝐷↑2) = 0)))
45	13, 44	mpd 16	. . 3 ⊢ (𝜑 → ¬ ((𝐶↑2) = 0 ∧ (𝐷↑2) = 0))
46	8	zcnd 12804	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47		sqeq0 14263	. . . . 5 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
48	46, 47	syl 18	. . . 4 ⊢ (𝜑 → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
49	12	zcnd 12804	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
50		sqeq0 14263	. . . . 5 ⊢ (𝐷 ∈ ℂ → ((𝐷↑2) = 0 ↔ 𝐷 = 0))
51	49, 50	syl 18	. . . 4 ⊢ (𝜑 → ((𝐷↑2) = 0 ↔ 𝐷 = 0))
52	48, 51	anbi12d 644	. . 3 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) ↔ (𝐶 = 0 ∧ 𝐷 = 0)))
53	45, 52	mtbid 327	. 2 ⊢ (𝜑 → ¬ (𝐶 = 0 ∧ 𝐷 = 0))
54		gcdn0cl 16672	. 2 ⊢ (((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ ¬ (𝐶 = 0 ∧ 𝐷 = 0)) → (𝐶 gcd 𝐷) ∈ ℕ)
55	8, 12, 53, 54	syl21anc 851	1 ⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {cab 2739 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 class class class wbr 5103 ↦ cmpt 5186 ran crn 5652 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 0cc0 11200 1c1 11201 + caddc 11203 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℕcn 12335 2c2 12397 ℤcz 12693 ℤ_≥cuz 12965 ...cfz 13639 mod cmo 14009 ↑cexp 14204 abscabs 15401 ∥ cdvds 16422 gcd cgcd 16664 ℤ[i]cgz 17107

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-sup 9434 df-inf 9435 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-rp 13121 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-dvds 16423 df-gcd 16665

This theorem is used by: 2sqlem8 27753

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