2sqlem8a - Metamath Proof Explorer

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Theorem 2sqlem8a 27394

Description: Lemma for 2sqlem8 27395. (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 12837	. . . . . 6 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
4	2, 3	sylib 218	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
5	4	simpld 494	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6		2sqlem8.c	. . . 4 ⊢ 𝐶 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
7	1, 5, 6	4sqlem5 16872	. . 3 ⊢ (𝜑 → (𝐶 ∈ ℤ ∧ ((𝐴 − 𝐶) / 𝑀) ∈ ℤ))
8	7	simpld 494	. 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 16872	. . 3 ⊢ (𝜑 → (𝐷 ∈ ℤ ∧ ((𝐵 − 𝐷) / 𝑀) ∈ ℤ))
12	11	simpld 494	. 2 ⊢ (𝜑 → 𝐷 ∈ ℤ)
13	4	simprd 495	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 1)
14		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝐶↑2) = 0)
15	1, 5, 6, 14	4sqlem9 16876	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝑀↑2) ∥ (𝐴↑2))
16	15	ex 412	. . . . . . . 8 ⊢ (𝜑 → ((𝐶↑2) = 0 → (𝑀↑2) ∥ (𝐴↑2)))
17		eluzelz 12763	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
18	2, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19		dvdssq 16496	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2)))
20	18, 1, 19	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2)))
21	16, 20	sylibrd 259	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑2) = 0 → 𝑀 ∥ 𝐴))
22		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝐷↑2) = 0)
23	9, 5, 10, 22	4sqlem9 16876	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝑀↑2) ∥ (𝐵↑2))
24	23	ex 412	. . . . . . . 8 ⊢ (𝜑 → ((𝐷↑2) = 0 → (𝑀↑2) ∥ (𝐵↑2)))
25		dvdssq 16496	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2)))
26	18, 9, 25	syl2anc 584	. . . . . . . 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 11097	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≠ 0)
31	28, 30	eqnetrd 2999	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≠ 0)
32	31	neneqd 2937	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴 gcd 𝐵) = 0)
33		gcdeq0 16446	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
34	1, 9, 33	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
35	32, 34	mtbid 324	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
36		dvdslegcd 16433	. . . . . . . 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 1375	. . . . . . 7 ⊢ (𝜑 → ((𝑀 ∥ 𝐴 ∧ 𝑀 ∥ 𝐵) → 𝑀 ≤ (𝐴 gcd 𝐵)))
38	21, 27, 37	syl2and 608	. . . . . 6 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 ≤ (𝐴 gcd 𝐵)))
39	28	breq2d 5110	. . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 ≤ 1))
40		nnle1eq1 12177	. . . . . . . 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 239	. . . . 5 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 = 1))
44	43	necon3ad 2945	. . . 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 12599	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47		sqeq0 14045	. . . . 5 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
48	46, 47	syl 17	. . . 4 ⊢ (𝜑 → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
49	12	zcnd 12599	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
50		sqeq0 14045	. . . . 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 16431	. 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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {cab 2714 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 class class class wbr 5098 ↦ cmpt 5179 ran crn 5625 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 0cc0 11028 1c1 11029 + caddc 11031 ≤ cle 11169 − cmin 11366 / cdiv 11796 ℕcn 12147 2c2 12202 ℤcz 12490 ℤ_≥cuz 12753 ...cfz 13425 mod cmo 13791 ↑cexp 13986 abscabs 15159 ∥ cdvds 16181 gcd cgcd 16423 ℤ[i]cgz 16859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-rp 12908 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-dvds 16182 df-gcd 16424

This theorem is referenced by: 2sqlem8 27395

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