Theorem 2sqlem7 27663

Description: Lemma for 2sq 27669. (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 3100	. . . . . 6 ⊢ (∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
4	3	reximi 3100	. . . . 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 27657	. . . . 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 11196	. . . . . . . . . 10 ⊢ 1 ≠ 0
9		gcdeq0 16610	. . . . . . . . . . . . 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 2762	. . . . . . . . . . . 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 2992	. . . . . . . . . 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 14205	. . . . . . . . . . . . 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 12593	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℝ)
19	17	nn0ge0d 12595	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑥↑2))
20		zsqcl2 14205	. . . . . . . . . . . . 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 12593	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℝ)
23	21	nn0ge0d 12595	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑦↑2))
24		add20 11753	. . . . . . . . . . 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 12623	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
27	26	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑥 ∈ ℂ)
28		zcn 12623	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
29	28	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑦 ∈ ℂ)
30		sqeq0 14187	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0))
31		sqeq0 14187	. . . . . . . . . . . 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 12566	. . . . . . . . . . . 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 12533	. . . . . . . . . 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 2848	. . . . . . 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 3204	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ)
47	7, 46	elind 4146	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ (𝑆 ∩ ℕ))
48	47	abssi 4016	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⊆ (𝑆 ∩ ℕ)
49	1, 48	eqsstri 3977	1 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∃wrex 3086   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ran crn 5656  ‘cfv 6533  (class class class)co 7414  ℂcc 11125  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130   ≤ cle 11271  ℕcn 12260  2c2 12322  ℕ₀cn0 12531  ℤcz 12618  ↑cexp 14128  abscabs 15324   gcd cgcd 16587  ℤ[i]cgz 17024

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 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-inf 9416  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-rp 13046  df-seq 14069  df-exp 14129  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-dvds 16346  df-gcd 16588  df-gz 17025

This theorem is used by:  2sqlem8  27665  2sqlem9  27666