Theorem 2sqlem7 27808

Description: Lemma for 2sq 27814. (Contributed by Mario Carneiro, 19-Jun-2015.)

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)))}

Assertion

Ref	Expression
2sqlem7	⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑌,𝑦

Allowed substitution hints:   𝑆(𝑤)   𝑌(𝑧, 𝑤)

Proof of Theorem 2sqlem7

Step	Hyp	Ref	Expression
1		2sqlem7.2	. 2 ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}
2		simpr 490	. . . . . . 7 ⊢ (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 = ((𝑥↑2) + (𝑦↑2)))
3	2	reximi 3104	. . . . . 6 ⊢ (∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
4	3	reximi 3104	. . . . 5 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
5		2sq.1	. . . . . 6 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
6	5	2sqlem2 27802	. . . . 5 ⊢ (𝑧 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
7	4, 6	sylibr 237	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ 𝑆)
8		ax-1ne0 11308	. . . . . . . . . 10 ⊢ 1 ≠ 0
9		gcdeq0 16733	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
10	9	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
11		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥 gcd 𝑦) = 1)
12	11	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ 1 = 0))
13	10, 12	bitr3d 284	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 = 0))
14	13	necon3bbid 2996	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (¬ (𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 ≠ 0))
15	8, 14	mpbiri 261	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ (𝑥 = 0 ∧ 𝑦 = 0))
16		zsqcl2 14321	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
17	16	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℕ0)
18	17	nn0red 12708	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℝ)
19	17	nn0ge0d 12710	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑥↑2))
20		zsqcl2 14321	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ0)
21	20	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℕ0)
22	21	nn0red 12708	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℝ)
23	21	nn0ge0d 12710	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑦↑2))
24		add20 11868	. . . . . . . . . . 11 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ ((𝑦↑2) ∈ ℝ ∧ 0 ≤ (𝑦↑2))) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ ((𝑥↑2) = 0 ∧ (𝑦↑2) = 0)))
25	18, 19, 22, 23, 24	syl22anc 852	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ ((𝑥↑2) = 0 ∧ (𝑦↑2) = 0)))
26		zcn 12738	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
27	26	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑥 ∈ ℂ)
28		zcn 12738	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
29	28	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑦 ∈ ℂ)
30		sqeq0 14303	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0))
31		sqeq0 14303	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
32	30, 31	bi2anan9 650	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
33	27, 29, 32	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
34	25, 33	bitrd 282	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
35	15, 34	mtbird 328	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ ((𝑥↑2) + (𝑦↑2)) = 0)
36		nn0addcl 12681	. . . . . . . . . . . 12 ⊢ (((𝑥↑2) ∈ ℕ0 ∧ (𝑦↑2) ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
37	16, 20, 36	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
38	37	adantr 486	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
39		elnn0 12648	. . . . . . . . . 10 ⊢ (((𝑥↑2) + (𝑦↑2)) ∈ ℕ0 ↔ (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
40	38, 39	sylib 221	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
41	40	ord 878	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (¬ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ → ((𝑥↑2) + (𝑦↑2)) = 0))
42	35, 41	mt3d 149	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ)
43		eleq1 2852	. . . . . . 7 ⊢ (𝑧 = ((𝑥↑2) + (𝑦↑2)) → (𝑧 ∈ ℕ ↔ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ))
44	42, 43	syl5ibrcom 250	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑧 = ((𝑥↑2) + (𝑦↑2)) → 𝑧 ∈ ℕ))
45	44	expimpd 459	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ))
46	45	rexlimivv 3208	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ)
47	7, 46	elind 4100	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ (𝑆 ∩ ℕ))
48	47	abssi 3996	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⊆ (𝑆 ∩ ℕ)
49	1, 48	eqsstri 3959	1 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5108   ↦ cmpt 5192  ran crn 5659  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  ℝcr 11238  0cc0 11239  1c1 11240   + caddc 11242   ≤ cle 11383  ℕcn 12375  2c2 12437  ℕ₀cn0 12646  ℤcz 12733  ↑cexp 14244  abscabs 15444   gcd cgcd 16708  ℤ[i]cgz 17151

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-seq 14185  df-exp 14245  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-dvds 16467  df-gcd 16709  df-gz 17152

This theorem is used by:  2sqlem8  27810  2sqlem9  27811