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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmspecpos | Structured version Visualization version GIF version |
Description: The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
Ref | Expression |
---|---|
rmspecpos | ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelre 12832 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
2 | 1 | resqcld 14089 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
3 | 1red 11214 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
4 | 2, 3 | resubcld 11641 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℝ) |
5 | sq1 14158 | . . . 4 ⊢ (1↑2) = 1 | |
6 | eluz2b1 12902 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℤ ∧ 1 < 𝐴)) | |
7 | 6 | simprbi 497 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) |
8 | 0le1 11736 | . . . . . . 7 ⊢ 0 ≤ 1 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 1) |
10 | eluzge2nn0 12870 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ0) | |
11 | 10 | nn0ge0d 12534 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 𝐴) |
12 | 3, 1, 9, 11 | lt2sqd 14218 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < 𝐴 ↔ (1↑2) < (𝐴↑2))) |
13 | 7, 12 | mpbid 231 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1↑2) < (𝐴↑2)) |
14 | 5, 13 | eqbrtrrid 5184 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < (𝐴↑2)) |
15 | 3, 2 | posdifd 11800 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < (𝐴↑2) ↔ 0 < ((𝐴↑2) − 1))) |
16 | 14, 15 | mpbid 231 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 < ((𝐴↑2) − 1)) |
17 | 4, 16 | elrpd 13012 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 0cc0 11109 1c1 11110 < clt 11247 ≤ cle 11248 − cmin 11443 2c2 12266 ℤcz 12557 ℤ≥cuz 12821 ℝ+crp 12973 ↑cexp 14026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-seq 13966 df-exp 14027 |
This theorem is referenced by: rmxycomplete 41646 rmxy1 41651 rmxy0 41652 rmxypos 41676 jm2.23 41725 |
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