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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundne1 | Structured version Visualization version GIF version |
Description: The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
Ref | Expression |
---|---|
pellfundne1 | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 11254 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ∈ ℝ) | |
2 | pellfundgt1 42575 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷)) | |
3 | 1, 2 | gtned 11388 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 ‘cfv 6544 1c1 11148 ℕcn 12256 ◻NNcsquarenn 42528 PellFundcpellfund 42532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-inf2 9675 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-omul 8491 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9476 df-inf 9477 df-oi 9544 df-card 9973 df-acn 9976 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-n0 12517 df-xnn0 12589 df-z 12603 df-uz 12867 df-q 12977 df-rp 13021 df-ico 13376 df-fz 13531 df-fl 13804 df-mod 13882 df-seq 14014 df-exp 14074 df-hash 14341 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-dvds 16250 df-gcd 16488 df-numer 16730 df-denom 16731 df-squarenn 42533 df-pell1qr 42534 df-pell14qr 42535 df-pell1234qr 42536 df-pellfund 42537 |
This theorem is referenced by: pellfund14 42590 |
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