Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pell1234qrre | Structured version Visualization version GIF version |
Description: General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
Ref | Expression |
---|---|
pell1234qrre | ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpell1234qr 37415 | . 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 653 | 1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∖ cdif 3571 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 ℕcn 11020 2c2 11070 ℤcz 11377 ↑cexp 12860 √csqrt 13973 ◻NNcsquarenn 37400 Pell1234QRcpell1234qr 37402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-pell1234qr 37408 |
This theorem is referenced by: pell1234qrreccl 37418 pell14qrre 37421 elpell14qr2 37426 pell14qrmulcl 37427 pell14qrreccl 37428 |
Copyright terms: Public domain | W3C validator |