MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2sqlem7 Structured version   Visualization version   GIF version

Theorem 2sqlem7 26927
Description: Lemma for 2sq 26933. (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
StepHypRef Expression
1 2sqlem7.2 . 2 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}
2 simpr 486 . . . . . . 7 (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 = ((𝑥↑2) + (𝑦↑2)))
32reximi 3085 . . . . . 6 (∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
43reximi 3085 . . . . 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))
652sqlem2 26921 . . . . 5 (𝑧𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝑥↑2) + (𝑦↑2)))
74, 6sylibr 233 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧𝑆)
8 ax-1ne0 11179 . . . . . . . . . 10 1 ≠ 0
9 gcdeq0 16458 . . . . . . . . . . . . 13 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
109adantr 482 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
11 simpr 486 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥 gcd 𝑦) = 1)
1211eqeq1d 2735 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 gcd 𝑦) = 0 ↔ 1 = 0))
1310, 12bitr3d 281 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 = 0))
1413necon3bbid 2979 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (¬ (𝑥 = 0 ∧ 𝑦 = 0) ↔ 1 ≠ 0))
158, 14mpbiri 258 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ (𝑥 = 0 ∧ 𝑦 = 0))
16 zsqcl2 14103 . . . . . . . . . . . . 13 (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
1716ad2antrr 725 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℕ0)
1817nn0red 12533 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑥↑2) ∈ ℝ)
1917nn0ge0d 12535 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑥↑2))
20 zsqcl2 14103 . . . . . . . . . . . . 13 (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ0)
2120ad2antlr 726 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℕ0)
2221nn0red 12533 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑦↑2) ∈ ℝ)
2321nn0ge0d 12535 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 0 ≤ (𝑦↑2))
24 add20 11726 . . . . . . . . . . 11 ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ ((𝑦↑2) ∈ ℝ ∧ 0 ≤ (𝑦↑2))) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ ((𝑥↑2) = 0 ∧ (𝑦↑2) = 0)))
2518, 19, 22, 23, 24syl22anc 838 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ ((𝑥↑2) = 0 ∧ (𝑦↑2) = 0)))
26 zcn 12563 . . . . . . . . . . . 12 (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
2726ad2antrr 725 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑥 ∈ ℂ)
28 zcn 12563 . . . . . . . . . . . 12 (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
2928ad2antlr 726 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → 𝑦 ∈ ℂ)
30 sqeq0 14085 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0))
31 sqeq0 14085 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
3230, 31bi2anan9 638 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
3327, 29, 32syl2anc 585 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) = 0 ∧ (𝑦↑2) = 0) ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
3425, 33bitrd 279 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
3515, 34mtbird 325 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ¬ ((𝑥↑2) + (𝑦↑2)) = 0)
36 nn0addcl 12507 . . . . . . . . . . . 12 (((𝑥↑2) ∈ ℕ0 ∧ (𝑦↑2) ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
3716, 20, 36syl2an 597 . . . . . . . . . . 11 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
3837adantr 482 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
39 elnn0 12474 . . . . . . . . . 10 (((𝑥↑2) + (𝑦↑2)) ∈ ℕ0 ↔ (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
4038, 39sylib 217 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (((𝑥↑2) + (𝑦↑2)) ∈ ℕ ∨ ((𝑥↑2) + (𝑦↑2)) = 0))
4140ord 863 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (¬ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ → ((𝑥↑2) + (𝑦↑2)) = 0))
4235, 41mt3d 148 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ)
43 eleq1 2822 . . . . . . 7 (𝑧 = ((𝑥↑2) + (𝑦↑2)) → (𝑧 ∈ ℕ ↔ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ))
4442, 43syl5ibrcom 246 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 gcd 𝑦) = 1) → (𝑧 = ((𝑥↑2) + (𝑦↑2)) → 𝑧 ∈ ℕ))
4544expimpd 455 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ))
4645rexlimivv 3200 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ ℕ)
477, 46elind 4195 . . 3 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) → 𝑧 ∈ (𝑆 ∩ ℕ))
4847abssi 4068 . 2 {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))} ⊆ (𝑆 ∩ ℕ)
491, 48eqsstri 4017 1 𝑌 ⊆ (𝑆 ∩ ℕ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  {cab 2710  wne 2941  wrex 3071  cin 3948  wss 3949   class class class wbr 5149  cmpt 5232  ran crn 5678  cfv 6544  (class class class)co 7409  cc 11108  cr 11109  0cc0 11110  1c1 11111   + caddc 11113  cle 11249  cn 12212  2c2 12267  0cn0 12472  cz 12558  cexp 14027  abscabs 15181   gcd cgcd 16435  ℤ[i]cgz 16862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-dvds 16198  df-gcd 16436  df-gz 16863
This theorem is referenced by:  2sqlem8  26929  2sqlem9  26930
  Copyright terms: Public domain W3C validator