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Mirrors > Home > MPE Home > Th. List > 2lgslem3a1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 2lgslem3 25907. (Contributed by AV, 15-Jul-2021.) |
Ref | Expression |
---|---|
2lgslem2.n | ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) |
Ref | Expression |
---|---|
2lgslem3a1 | ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11892 | . . . 4 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
2 | 8nn 11720 | . . . . 5 ⊢ 8 ∈ ℕ | |
3 | nnrp 12388 | . . . . 5 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 8 ∈ ℝ+ |
5 | modmuladdnn0 13271 | . . . 4 ⊢ ((𝑃 ∈ ℕ0 ∧ 8 ∈ ℝ+) → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) | |
6 | 1, 4, 5 | sylancl 586 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) |
7 | simpr 485 | . . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
8 | nn0cn 11895 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
9 | 8cn 11722 | . . . . . . . . . . . 12 ⊢ 8 ∈ ℂ | |
10 | 9 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 8 ∈ ℂ) |
11 | 8, 10 | mulcomd 10650 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑘 · 8) = (8 · 𝑘)) |
12 | 11 | adantl 482 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑘 · 8) = (8 · 𝑘)) |
13 | 12 | oveq1d 7160 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑘 · 8) + 1) = ((8 · 𝑘) + 1)) |
14 | 13 | eqeq2d 2829 | . . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 1) ↔ 𝑃 = ((8 · 𝑘) + 1))) |
15 | 14 | biimpa 477 | . . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑃 = ((8 · 𝑘) + 1)) |
16 | 2lgslem2.n | . . . . . . 7 ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) | |
17 | 16 | 2lgslem3a 25899 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝑘) + 1)) → 𝑁 = (2 · 𝑘)) |
18 | 7, 15, 17 | syl2an2r 681 | . . . . 5 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑁 = (2 · 𝑘)) |
19 | oveq1 7152 | . . . . . 6 ⊢ (𝑁 = (2 · 𝑘) → (𝑁 mod 2) = ((2 · 𝑘) mod 2)) | |
20 | 2cnd 11703 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 2 ∈ ℂ) | |
21 | 20, 8 | mulcomd 10650 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2 · 𝑘) = (𝑘 · 2)) |
22 | 21 | oveq1d 7160 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = ((𝑘 · 2) mod 2)) |
23 | nn0z 11993 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
24 | 2rp 12382 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
25 | mulmod0 13233 | . . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ 2 ∈ ℝ+) → ((𝑘 · 2) mod 2) = 0) | |
26 | 23, 24, 25 | sylancl 586 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 · 2) mod 2) = 0) |
27 | 22, 26 | eqtrd 2853 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = 0) |
28 | 19, 27 | sylan9eqr 2875 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑁 = (2 · 𝑘)) → (𝑁 mod 2) = 0) |
29 | 7, 18, 28 | syl2an2r 681 | . . . 4 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → (𝑁 mod 2) = 0) |
30 | 29 | rexlimdva2 3284 | . . 3 ⊢ (𝑃 ∈ ℕ → (∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1) → (𝑁 mod 2) = 0)) |
31 | 6, 30 | syld 47 | . 2 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → (𝑁 mod 2) = 0)) |
32 | 31 | imp 407 | 1 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 − cmin 10858 / cdiv 11285 ℕcn 11626 2c2 11680 4c4 11682 8c8 11686 ℕ0cn0 11885 ℤcz 11969 ℝ+crp 12377 ⌊cfl 13148 mod cmo 13225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fl 13150 df-mod 13226 |
This theorem is referenced by: 2lgslem3 25907 |
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