Theorem 2sqlem7 26809

Description: Lemma for 2sq 26815. (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 485	. . . . . . 7 ⊢ (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 = ((𝑥↑2) + (𝑦↑2)))
3	2	reximi 3083	. . . . . 6 ⊢ (∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
4	3	reximi 3083	. . . . 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 26803	. . . . 5 ⊢ (𝑧 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
7	4, 6	sylibr 233	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ 𝑆)
8		ax-1ne0 11129	. . . . . . . . . 10 ⊢ 1 ≠ 0
9		gcdeq0 16408	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
10	9	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
11		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥 gcd 𝑦) = 1)
12	11	eqeq1d 2733	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ 1 = 0))
13	10, 12	bitr3d 280	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 = 0))
14	13	necon3bbid 2977	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (¬ (𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 ≠ 0))
15	8, 14	mpbiri 257	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ (𝑥 = 0 ∧ 𝑦 = 0))
16		zsqcl2 14053	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
17	16	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℕ0)
18	17	nn0red 12483	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℝ)
19	17	nn0ge0d 12485	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑥↑2))
20		zsqcl2 14053	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ0)
21	20	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℕ0)
22	21	nn0red 12483	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℝ)
23	21	nn0ge0d 12485	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑦↑2))
24		add20 11676	. . . . . . . . . . 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 837	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ ((𝑥↑2) = 0 ∧ (𝑦↑2) = 0)))
26		zcn 12513	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
27	26	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑥 ∈ ℂ)
28		zcn 12513	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
29	28	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑦 ∈ ℂ)
30		sqeq0 14035	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0))
31		sqeq0 14035	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
32	30, 31	bi2anan9 637	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
33	27, 29, 32	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
34	25, 33	bitrd 278	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
35	15, 34	mtbird 324	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ ((𝑥↑2) + (𝑦↑2)) = 0)
36		nn0addcl 12457	. . . . . . . . . . . 12 ⊢ (((𝑥↑2) ∈ ℕ0 ∧ (𝑦↑2) ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
37	16, 20, 36	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
38	37	adantr 481	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
39		elnn0 12424	. . . . . . . . . 10 ⊢ (((𝑥↑2) + (𝑦↑2)) ∈ ℕ0 ↔ (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
40	38, 39	sylib 217	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
41	40	ord 862	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (¬ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ → ((𝑥↑2) + (𝑦↑2)) = 0))
42	35, 41	mt3d 148	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ)
43		eleq1 2820	. . . . . . 7 ⊢ (𝑧 = ((𝑥↑2) + (𝑦↑2)) → (𝑧 ∈ ℕ ↔ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ))
44	42, 43	syl5ibrcom 246	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑧 = ((𝑥↑2) + (𝑦↑2)) → 𝑧 ∈ ℕ))
45	44	expimpd 454	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ))
46	45	rexlimivv 3192	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ)
47	7, 46	elind 4159	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ (𝑆 ∩ ℕ))
48	47	abssi 4032	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⊆ (𝑆 ∩ ℕ)
49	1, 48	eqsstri 3981	1 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2708   ≠ wne 2939  ∃wrex 3069   ∩ cin 3912   ⊆ wss 3913   class class class wbr 5110   ↦ cmpt 5193  ran crn 5639  ‘cfv 6501  (class class class)co 7362  ℂcc 11058  ℝcr 11059  0cc0 11060  1c1 11061   + caddc 11063   ≤ cle 11199  ℕcn 12162  2c2 12217  ℕ₀cn0 12422  ℤcz 12508  ↑cexp 13977  abscabs 15131   gcd cgcd 16385  ℤ[i]cgz 16812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9387  df-inf 9388  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-seq 13917  df-exp 13978  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-dvds 16148  df-gcd 16386  df-gz 16813

This theorem is referenced by:  2sqlem8  26811  2sqlem9  26812