2sqlem8a - Metamath Proof Explorer

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Theorem 2sqlem8a 27351

Description: Lemma for 2sqlem8 27352. (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 12930	. . . . . 6 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
4	2, 3	sylib 217	. . . . 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 16904	. . 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 16904	. . 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 16908	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝑀↑2) ∥ (𝐴↑2))
16	15	ex 412	. . . . . . . 8 ⊢ (𝜑 → ((𝐶↑2) = 0 → (𝑀↑2) ∥ (𝐴↑2)))
17		eluzelz 12856	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
18	2, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19		dvdssq 16531	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2)))
20	18, 1, 19	syl2anc 583	. . . . . . . 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 16908	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝑀↑2) ∥ (𝐵↑2))
24	23	ex 412	. . . . . . . 8 ⊢ (𝜑 → ((𝐷↑2) = 0 → (𝑀↑2) ∥ (𝐵↑2)))
25		dvdssq 16531	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑀 ∥ 𝐵 ↔ (𝑀↑2) ∥ (𝐵↑2)))
26	18, 9, 25	syl2anc 583	. . . . . . . 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 11201	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≠ 0)
31	28, 30	eqnetrd 3003	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≠ 0)
32	31	neneqd 2940	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴 gcd 𝐵) = 0)
33		gcdeq0 16485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
34	1, 9, 33	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
35	32, 34	mtbid 324	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
36		dvdslegcd 16472	. . . . . . . 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 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑀 ∥ 𝐴 ∧ 𝑀 ∥ 𝐵) → 𝑀 ≤ (𝐴 gcd 𝐵)))
38	21, 27, 37	syl2and 607	. . . . . 6 ⊢ (𝜑 → (((𝐶↑2) = 0 ∧ (𝐷↑2) = 0) → 𝑀 ≤ (𝐴 gcd 𝐵)))
39	28	breq2d 5154	. . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 ≤ 1))
40		nnle1eq1 12266	. . . . . . . 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 2948	. . . 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 12691	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47		sqeq0 14110	. . . . 5 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
48	46, 47	syl 17	. . . 4 ⊢ (𝜑 → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
49	12	zcnd 12691	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
50		sqeq0 14110	. . . . 5 ⊢ (𝐷 ∈ ℂ → ((𝐷↑2) = 0 ↔ 𝐷 = 0))
51	49, 50	syl 17	. . . 4 ⊢ (𝜑 → ((𝐷↑2) = 0 ↔ 𝐷 = 0))
52	48, 51	anbi12d 630	. . 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 16470	. 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 395 = wceq 1534 ∈ wcel 2099 {cab 2704 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 class class class wbr 5142 ↦ cmpt 5225 ran crn 5673 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 0cc0 11132 1c1 11133 + caddc 11135 ≤ cle 11273 − cmin 11468 / cdiv 11895 ℕcn 12236 2c2 12291 ℤcz 12582 ℤ_≥cuz 12846 ...cfz 13510 mod cmo 13860 ↑cexp 14052 abscabs 15207 ∥ cdvds 16224 gcd cgcd 16462 ℤ[i]cgz 16891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16225 df-gcd 16463

This theorem is referenced by: 2sqlem8 27352

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