Theorem 2sqlem7 16445

Description: Lemma for 2sq . (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 110	. . . . . . 7 ⊢ (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 = ((𝑥↑2) + (𝑦↑2)))
3	2	reximi 2647	. . . . . 6 ⊢ (∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
4	3	reximi 2647	. . . . 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 16439	. . . . 5 ⊢ (𝑧 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
7	4, 6	sylibr 134	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ 𝑆)
8		1ne0 9381	. . . . . . . . . 10 ⊢ 1 ≠ 0
9		gcdeq0 12779	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
10	9	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
11		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥 gcd 𝑦) = 1)
12	11	eqeq1d 2247	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ 1 = 0))
13	10, 12	bitr3d 190	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 = 0))
14	13	necon3bbid 2460	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (¬ (𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 ≠ 0))
15	8, 14	mpbiri 168	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ (𝑥 = 0 ∧ 𝑦 = 0))
16		zsqcl2 11075	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
17	16	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℕ0)
18	17	nn0red 9632	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℝ)
19	17	nn0ge0d 9634	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑥↑2))
20		zsqcl2 11075	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ0)
21	20	ad2antlr 493	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℕ0)
22	21	nn0red 9632	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℝ)
23	21	nn0ge0d 9634	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑦↑2))
24		add20 8809	. . . . . . . . . . 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 1279	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ ((𝑥↑2) = 0 ∧ (𝑦↑2) = 0)))
26		zcn 9660	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
27	26	ad2antrr 492	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑥 ∈ ℂ)
28		zcn 9660	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
29	28	ad2antlr 493	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑦 ∈ ℂ)
30		sqeq0 11060	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0))
31		sqeq0 11060	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
32	30, 31	bi2anan9 614	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
33	27, 29, 32	syl2anc 415	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
34	25, 33	bitrd 188	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
35	15, 34	mtbird 684	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ ((𝑥↑2) + (𝑦↑2)) = 0)
36		nn0addcl 9609	. . . . . . . . . . 11 ⊢ (((𝑥↑2) ∈ ℕ0 ∧ (𝑦↑2) ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
37	16, 20, 36	syl2an 289	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
38	37	adantr 276	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
39		elnn0 9576	. . . . . . . . 9 ⊢ (((𝑥↑2) + (𝑦↑2)) ∈ ℕ0 ↔ (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
40	38, 39	sylib 122	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
41	35, 40	ecased 1390	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ)
42		eleq1 2301	. . . . . . 7 ⊢ (𝑧 = ((𝑥↑2) + (𝑦↑2)) → (𝑧 ∈ ℕ ↔ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ))
43	41, 42	syl5ibrcom 157	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑧 = ((𝑥↑2) + (𝑦↑2)) → 𝑧 ∈ ℕ))
44	43	expimpd 363	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ))
45	44	rexlimivv 2674	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ)
46	7, 45	elind 3414	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ (𝑆 ∩ ℕ))
47	46	abssi 3323	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⊆ (𝑆 ∩ ℕ)
48	1, 47	eqsstri 3280	1 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  {cab 2224   ≠ wne 2420  ∃wrex 2529   ∩ cin 3219   ⊆ wss 3220   class class class wbr 4131   ↦ cmpt 4193  ran crn 4777  ‘cfv 5380  (class class class)co 6088  ℂcc 8183  ℝcr 8184  0cc0 8185  1c1 8186   + caddc 8188   ≤ cle 8367  ℕcn 9313  2c2 9364  ℕ₀cn0 9574  ℤcz 9655  ↑cexp 10996  abscabs 11785   gcd cgcd 12755  ℤ[i]cgz 13177

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-gcd 12756  df-gz 13178

This theorem is used by:  2sqlem8  16447  2sqlem9  16448