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Mirrors > Home > ILE Home > Th. List > bernneq3 | GIF version |
Description: A corollary of bernneq 10380. (Contributed by Mario Carneiro, 11-Mar-2014.) |
Ref | Expression |
---|---|
bernneq3 | ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 8954 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | 1 | adantl 275 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℝ) |
3 | peano2re 7866 | . . 3 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
4 | 2, 3 | syl 14 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℝ) |
5 | eluzelre 9304 | . . 3 ⊢ (𝑃 ∈ (ℤ≥‘2) → 𝑃 ∈ ℝ) | |
6 | reexpcl 10278 | . . 3 ⊢ ((𝑃 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑃↑𝑁) ∈ ℝ) | |
7 | 5, 6 | sylan 281 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑃↑𝑁) ∈ ℝ) |
8 | 2 | ltp1d 8656 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑁 + 1)) |
9 | uz2m1nn 9367 | . . . . . . 7 ⊢ (𝑃 ∈ (ℤ≥‘2) → (𝑃 − 1) ∈ ℕ) | |
10 | 9 | adantr 274 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑃 − 1) ∈ ℕ) |
11 | 10 | nnred 8701 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑃 − 1) ∈ ℝ) |
12 | 11, 2 | remulcld 7764 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → ((𝑃 − 1) · 𝑁) ∈ ℝ) |
13 | peano2re 7866 | . . . 4 ⊢ (((𝑃 − 1) · 𝑁) ∈ ℝ → (((𝑃 − 1) · 𝑁) + 1) ∈ ℝ) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (((𝑃 − 1) · 𝑁) + 1) ∈ ℝ) |
15 | 1red 7749 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 1 ∈ ℝ) | |
16 | nn0ge0 8970 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
17 | 16 | adantl 275 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 0 ≤ 𝑁) |
18 | 10 | nnge1d 8731 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 1 ≤ (𝑃 − 1)) |
19 | 2, 11, 17, 18 | lemulge12d 8664 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ ((𝑃 − 1) · 𝑁)) |
20 | 2, 12, 15, 19 | leadd1dd 8289 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≤ (((𝑃 − 1) · 𝑁) + 1)) |
21 | 5 | adantr 274 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑃 ∈ ℝ) |
22 | simpr 109 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
23 | eluzge2nn0 9333 | . . . . . 6 ⊢ (𝑃 ∈ (ℤ≥‘2) → 𝑃 ∈ ℕ0) | |
24 | nn0ge0 8970 | . . . . . 6 ⊢ (𝑃 ∈ ℕ0 → 0 ≤ 𝑃) | |
25 | 23, 24 | syl 14 | . . . . 5 ⊢ (𝑃 ∈ (ℤ≥‘2) → 0 ≤ 𝑃) |
26 | 25 | adantr 274 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 0 ≤ 𝑃) |
27 | bernneq2 10381 | . . . 4 ⊢ ((𝑃 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑃) → (((𝑃 − 1) · 𝑁) + 1) ≤ (𝑃↑𝑁)) | |
28 | 21, 22, 26, 27 | syl3anc 1201 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (((𝑃 − 1) · 𝑁) + 1) ≤ (𝑃↑𝑁)) |
29 | 4, 14, 7, 20, 28 | letrd 7854 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≤ (𝑃↑𝑁)) |
30 | 2, 4, 7, 8, 29 | ltletrd 8153 | 1 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃↑𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 ℝcr 7587 0cc0 7588 1c1 7589 + caddc 7591 · cmul 7593 < clt 7768 ≤ cle 7769 − cmin 7901 ℕcn 8688 2c2 8739 ℕ0cn0 8945 ℤ≥cuz 9294 ↑cexp 10260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 df-z 9023 df-uz 9295 df-seqfrec 10187 df-exp 10261 |
This theorem is referenced by: resqrexlemcvg 10759 resqrexlemga 10763 pw2dvds 11771 cvgcmp2nlemabs 13154 trilpolemlt1 13161 |
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