2sqlem8a - Metamath Proof Explorer

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Theorem 2sqlem8a 27313

Description: Lemma for 2sqlem8 27314. (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 12910	. . . . . 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 16884	. . 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 16884	. . 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 16888	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶↑2) = 0) → (𝑀↑2) ∥ (𝐴↑2))
16	15	ex 412	. . . . . . . 8 ⊢ (𝜑 → ((𝐶↑2) = 0 → (𝑀↑2) ∥ (𝐴↑2)))
17		eluzelz 12836	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
18	2, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19		dvdssq 16511	. . . . . . . . 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 16888	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐷↑2) = 0) → (𝑀↑2) ∥ (𝐵↑2))
24	23	ex 412	. . . . . . . 8 ⊢ (𝜑 → ((𝐷↑2) = 0 → (𝑀↑2) ∥ (𝐵↑2)))
25		dvdssq 16511	. . . . . . . . 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 11181	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≠ 0)
31	28, 30	eqnetrd 3002	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≠ 0)
32	31	neneqd 2939	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴 gcd 𝐵) = 0)
33		gcdeq0 16465	. . . . . . . . . 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 16452	. . . . . . . 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 1370	. . . . . . 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 5153	. . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ (𝐴 gcd 𝐵) ↔ 𝑀 ≤ 1))
40		nnle1eq1 12246	. . . . . . . 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 2947	. . . 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 12671	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47		sqeq0 14090	. . . . 5 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
48	46, 47	syl 17	. . . 4 ⊢ (𝜑 → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
49	12	zcnd 12671	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
50		sqeq0 14090	. . . . 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 16450	. 2 ⊢ (((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ ¬ (𝐶 = 0 ∧ 𝐷 = 0)) → (𝐶 gcd 𝐷) ∈ ℕ)
55	8, 12, 53, 54	syl21anc 835	1 ⊢ (𝜑 → (𝐶 gcd 𝐷) ∈ ℕ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 class class class wbr 5141 ↦ cmpt 5224 ran crn 5670 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 ≤ cle 11253 − cmin 11448 / cdiv 11875 ℕcn 12216 2c2 12271 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13490 mod cmo 13840 ↑cexp 14032 abscabs 15187 ∥ cdvds 16204 gcd cgcd 16442 ℤ[i]cgz 16871

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443

This theorem is referenced by: 2sqlem8 27314

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