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Mirrors > Home > MPE Home > Th. List > bernneq3 | Structured version Visualization version GIF version |
Description: A corollary of bernneq 13584. (Contributed by Mario Carneiro, 11-Mar-2014.) |
Ref | Expression |
---|---|
bernneq3 | ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 11900 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℝ) |
3 | peano2re 10807 | . . 3 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℝ) |
5 | eluzelre 12248 | . . 3 ⊢ (𝑃 ∈ (ℤ≥‘2) → 𝑃 ∈ ℝ) | |
6 | reexpcl 13440 | . . 3 ⊢ ((𝑃 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑃↑𝑁) ∈ ℝ) | |
7 | 5, 6 | sylan 582 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑃↑𝑁) ∈ ℝ) |
8 | 2 | ltp1d 11564 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑁 + 1)) |
9 | uz2m1nn 12317 | . . . . . . 7 ⊢ (𝑃 ∈ (ℤ≥‘2) → (𝑃 − 1) ∈ ℕ) | |
10 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑃 − 1) ∈ ℕ) |
11 | 10 | nnred 11647 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑃 − 1) ∈ ℝ) |
12 | 11, 2 | remulcld 10665 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → ((𝑃 − 1) · 𝑁) ∈ ℝ) |
13 | peano2re 10807 | . . . 4 ⊢ (((𝑃 − 1) · 𝑁) ∈ ℝ → (((𝑃 − 1) · 𝑁) + 1) ∈ ℝ) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (((𝑃 − 1) · 𝑁) + 1) ∈ ℝ) |
15 | 1red 10636 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 1 ∈ ℝ) | |
16 | nn0ge0 11916 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
17 | 16 | adantl 484 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 0 ≤ 𝑁) |
18 | 10 | nnge1d 11679 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 1 ≤ (𝑃 − 1)) |
19 | 2, 11, 17, 18 | lemulge12d 11572 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ ((𝑃 − 1) · 𝑁)) |
20 | 2, 12, 15, 19 | leadd1dd 11248 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≤ (((𝑃 − 1) · 𝑁) + 1)) |
21 | 5 | adantr 483 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑃 ∈ ℝ) |
22 | simpr 487 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
23 | eluzge2nn0 12281 | . . . . . 6 ⊢ (𝑃 ∈ (ℤ≥‘2) → 𝑃 ∈ ℕ0) | |
24 | nn0ge0 11916 | . . . . . 6 ⊢ (𝑃 ∈ ℕ0 → 0 ≤ 𝑃) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝑃 ∈ (ℤ≥‘2) → 0 ≤ 𝑃) |
26 | 25 | adantr 483 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 0 ≤ 𝑃) |
27 | bernneq2 13585 | . . . 4 ⊢ ((𝑃 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑃) → (((𝑃 − 1) · 𝑁) + 1) ≤ (𝑃↑𝑁)) | |
28 | 21, 22, 26, 27 | syl3anc 1367 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (((𝑃 − 1) · 𝑁) + 1) ≤ (𝑃↑𝑁)) |
29 | 4, 14, 7, 20, 28 | letrd 10791 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≤ (𝑃↑𝑁)) |
30 | 2, 4, 7, 8, 29 | ltletrd 10794 | 1 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ℤ≥cuz 12237 ↑cexp 13423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-exp 13424 |
This theorem is referenced by: climcnds 15200 bitsfzo 15778 bitsinv1 15785 pcfaclem 16228 pcfac 16229 chpchtsum 25789 bposlem1 25854 |
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