Theorem pell1234qrre 42414

Description: General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.)

Assertion

Ref	Expression
pell1234qrre	⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)

Proof of Theorem pell1234qrre

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpell1234qr 42413	. 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1234QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1))))
2	1	simprbda 497	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059   ∖ cdif 3941  ‘cfv 6549  (class class class)co 7419  ℝcr 11139  1c1 11141   + caddc 11143   · cmul 11145   − cmin 11476  ℕcn 12245  2c2 12300  ℤcz 12591  ↑cexp 14062  √csqrt 15216  ◻_NNcsquarenn 42398  Pell1234QRcpell1234qr 42400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-cnex 11196  ax-resscn 11197

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-pell1234qr 42406

This theorem is referenced by:  pell1234qrreccl  42416  pell14qrre  42419  elpell14qr2  42424  pell14qrmulcl  42425  pell14qrreccl  42426