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Mirrors > Home > MPE Home > Th. List > 2lgslem3a1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 2lgslem3 27241. (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 12475 | . . . 4 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
2 | 8nn 12303 | . . . . 5 ⊢ 8 ∈ ℕ | |
3 | nnrp 12981 | . . . . 5 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 8 ∈ ℝ+ |
5 | modmuladdnn0 13876 | . . . 4 ⊢ ((𝑃 ∈ ℕ0 ∧ 8 ∈ ℝ+) → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) | |
6 | 1, 4, 5 | sylancl 585 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) |
7 | simpr 484 | . . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
8 | nn0cn 12478 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
9 | 8cn 12305 | . . . . . . . . . . . 12 ⊢ 8 ∈ ℂ | |
10 | 9 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 8 ∈ ℂ) |
11 | 8, 10 | mulcomd 11231 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑘 · 8) = (8 · 𝑘)) |
12 | 11 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑘 · 8) = (8 · 𝑘)) |
13 | 12 | oveq1d 7416 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑘 · 8) + 1) = ((8 · 𝑘) + 1)) |
14 | 13 | eqeq2d 2735 | . . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 1) ↔ 𝑃 = ((8 · 𝑘) + 1))) |
15 | 14 | biimpa 476 | . . . . . 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 27233 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝑘) + 1)) → 𝑁 = (2 · 𝑘)) |
18 | 7, 15, 17 | syl2an2r 682 | . . . . 5 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑁 = (2 · 𝑘)) |
19 | oveq1 7408 | . . . . . 6 ⊢ (𝑁 = (2 · 𝑘) → (𝑁 mod 2) = ((2 · 𝑘) mod 2)) | |
20 | 2cnd 12286 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 2 ∈ ℂ) | |
21 | 20, 8 | mulcomd 11231 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2 · 𝑘) = (𝑘 · 2)) |
22 | 21 | oveq1d 7416 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = ((𝑘 · 2) mod 2)) |
23 | nn0z 12579 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
24 | 2rp 12975 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
25 | mulmod0 13838 | . . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ 2 ∈ ℝ+) → ((𝑘 · 2) mod 2) = 0) | |
26 | 23, 24, 25 | sylancl 585 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 · 2) mod 2) = 0) |
27 | 22, 26 | eqtrd 2764 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = 0) |
28 | 19, 27 | sylan9eqr 2786 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑁 = (2 · 𝑘)) → (𝑁 mod 2) = 0) |
29 | 7, 18, 28 | syl2an2r 682 | . . . 4 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → (𝑁 mod 2) = 0) |
30 | 29 | rexlimdva2 3149 | . . 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 406 | 1 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 4c4 12265 8c8 12269 ℕ0cn0 12468 ℤcz 12554 ℝ+crp 12970 ⌊cfl 13751 mod cmo 13830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fl 13753 df-mod 13831 |
This theorem is referenced by: 2lgslem3 27241 |
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