Theorem pell1234qrre 43527

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 43526	. 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 503	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087   ∖ cdif 3901  ‘cfv 6536  (class class class)co 7410  ℝcr 11098  1c1 11100   + caddc 11102   · cmul 11104   − cmin 11440  ℕcn 12232  2c2 12294  ℤcz 12590  ↑cexp 14096  √csqrt 15283  ◻_NNcsquarenn 43511  Pell1234QRcpell1234qr 43513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404  ax-cnex 11155  ax-resscn 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-pell1234qr 43519

This theorem is referenced by:  pell1234qrreccl  43529  pell14qrre  43532  elpell14qr2  43537  pell14qrmulcl  43538  pell14qrreccl  43539