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Mirrors > Home > MPE Home > Th. List > Mathboxes > pell14qrdivcl | Structured version Visualization version GIF version |
Description: Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
Ref | Expression |
---|---|
pell14qrdivcl | ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 / 𝐵) ∈ (Pell14QR‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pell14qrre 40676 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 11004 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ) |
3 | 2 | 3adant3 1131 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ) |
4 | pell14qrre 40676 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → 𝐵 ∈ ℝ) | |
5 | 4 | recnd 11004 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → 𝐵 ∈ ℂ) |
6 | 5 | 3adant2 1130 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → 𝐵 ∈ ℂ) |
7 | pell14qrne0 40677 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → 𝐵 ≠ 0) | |
8 | 7 | 3adant2 1130 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → 𝐵 ≠ 0) |
9 | 3, 6, 8 | divrecd 11754 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
10 | pell14qrreccl 40683 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (1 / 𝐵) ∈ (Pell14QR‘𝐷)) | |
11 | 10 | 3adant2 1130 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (1 / 𝐵) ∈ (Pell14QR‘𝐷)) |
12 | pell14qrmulcl 40682 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 / 𝐵) ∈ (Pell14QR‘𝐷)) → (𝐴 · (1 / 𝐵)) ∈ (Pell14QR‘𝐷)) | |
13 | 11, 12 | syld3an3 1408 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 · (1 / 𝐵)) ∈ (Pell14QR‘𝐷)) |
14 | 9, 13 | eqeltrd 2841 | 1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 / 𝐵) ∈ (Pell14QR‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2110 ≠ wne 2945 ∖ cdif 3889 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 0cc0 10872 1c1 10873 · cmul 10877 / cdiv 11632 ℕcn 11973 ◻NNcsquarenn 40655 Pell14QRcpell14qr 40658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-rp 12730 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-pell14qr 40662 df-pell1234qr 40663 |
This theorem is referenced by: pell14qrexpclnn0 40685 pellfundex 40705 |
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