Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pell14qrre | Structured version Visualization version GIF version |
Description: A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
Ref | Expression |
---|---|
pell14qrre | ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pell14qrss1234 40409 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ (Pell1234QR‘𝐷)) | |
2 | 1 | sselda 3915 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ (Pell1234QR‘𝐷)) |
3 | pell1234qrre 40405 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ) | |
4 | 2, 3 | syldan 594 | 1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∖ cdif 3877 ‘cfv 6397 ℝcr 10752 ℕcn 11854 ◻NNcsquarenn 40389 Pell1234QRcpell1234qr 40391 Pell14QRcpell14qr 40392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-i2m1 10821 ax-1ne0 10822 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-ov 7234 df-om 7663 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-neg 11089 df-nn 11855 df-n0 12115 df-z 12201 df-pell14qr 40396 df-pell1234qr 40397 |
This theorem is referenced by: pell14qrrp 40413 pell14qrdivcl 40418 pell14qrexpclnn0 40419 pell14qrexpcl 40420 pell14qrdich 40422 elpell1qr2 40425 pell14qrgap 40428 pell14qrgapw 40429 pellfundre 40434 pellfundge 40435 pellfundlb 40437 pellfundglb 40438 pellfundex 40439 pellfund14gap 40440 pellfund14 40451 |
Copyright terms: Public domain | W3C validator |