![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2lgslem3a1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 2lgslem3 25328. (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 11491 | . . . 4 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
2 | 8nn 11383 | . . . . 5 ⊢ 8 ∈ ℕ | |
3 | nnrp 12035 | . . . . 5 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 8 ∈ ℝ+ |
5 | modmuladdnn0 12908 | . . . 4 ⊢ ((𝑃 ∈ ℕ0 ∧ 8 ∈ ℝ+) → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) | |
6 | 1, 4, 5 | sylancl 697 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) |
7 | simpr 479 | . . . . . 6 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
8 | nn0cn 11494 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
9 | 8cn 11298 | . . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ | |
10 | 9 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → 8 ∈ ℂ) |
11 | 8, 10 | mulcomd 10253 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (𝑘 · 8) = (8 · 𝑘)) |
12 | 11 | adantl 473 | . . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑘 · 8) = (8 · 𝑘)) |
13 | 12 | oveq1d 6828 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑘 · 8) + 1) = ((8 · 𝑘) + 1)) |
14 | 13 | eqeq2d 2770 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 1) ↔ 𝑃 = ((8 · 𝑘) + 1))) |
15 | 14 | biimpa 502 | . . . . . . 7 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑃 = ((8 · 𝑘) + 1)) |
16 | 2lgslem2.n | . . . . . . . 8 ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) | |
17 | 16 | 2lgslem3a 25320 | . . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝑘) + 1)) → 𝑁 = (2 · 𝑘)) |
18 | 7, 15, 17 | syl2an2r 911 | . . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑁 = (2 · 𝑘)) |
19 | oveq1 6820 | . . . . . . 7 ⊢ (𝑁 = (2 · 𝑘) → (𝑁 mod 2) = ((2 · 𝑘) mod 2)) | |
20 | 2cnd 11285 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 2 ∈ ℂ) | |
21 | 20, 8 | mulcomd 10253 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (2 · 𝑘) = (𝑘 · 2)) |
22 | 21 | oveq1d 6828 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = ((𝑘 · 2) mod 2)) |
23 | nn0z 11592 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
24 | 2rp 12030 | . . . . . . . . 9 ⊢ 2 ∈ ℝ+ | |
25 | mulmod0 12870 | . . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ 2 ∈ ℝ+) → ((𝑘 · 2) mod 2) = 0) | |
26 | 23, 24, 25 | sylancl 697 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 · 2) mod 2) = 0) |
27 | 22, 26 | eqtrd 2794 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = 0) |
28 | 19, 27 | sylan9eqr 2816 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑁 = (2 · 𝑘)) → (𝑁 mod 2) = 0) |
29 | 7, 18, 28 | syl2an2r 911 | . . . . 5 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → (𝑁 mod 2) = 0) |
30 | 29 | ex 449 | . . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 1) → (𝑁 mod 2) = 0)) |
31 | 30 | rexlimdva 3169 | . . 3 ⊢ (𝑃 ∈ ℕ → (∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1) → (𝑁 mod 2) = 0)) |
32 | 6, 31 | syld 47 | . 2 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → (𝑁 mod 2) = 0)) |
33 | 32 | imp 444 | 1 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∃wrex 3051 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 0cc0 10128 1c1 10129 + caddc 10131 · cmul 10133 − cmin 10458 / cdiv 10876 ℕcn 11212 2c2 11262 4c4 11264 8c8 11268 ℕ0cn0 11484 ℤcz 11569 ℝ+crp 12025 ⌊cfl 12785 mod cmo 12862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-ico 12374 df-fl 12787 df-mod 12863 |
This theorem is referenced by: 2lgslem3 25328 |
Copyright terms: Public domain | W3C validator |