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Mirrors > Home > MPE Home > Th. List > 2lgslem3b1 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgslem3 25974. (Contributed by AV, 16-Jul-2021.) |
Ref | Expression |
---|---|
2lgslem2.n | ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) |
Ref | Expression |
---|---|
2lgslem3b1 | ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 3) → (𝑁 mod 2) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11898 | . . . 4 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
2 | 8nn 11726 | . . . . 5 ⊢ 8 ∈ ℕ | |
3 | nnrp 12394 | . . . . 5 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 8 ∈ ℝ+ |
5 | modmuladdnn0 13277 | . . . 4 ⊢ ((𝑃 ∈ ℕ0 ∧ 8 ∈ ℝ+) → ((𝑃 mod 8) = 3 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 3))) | |
6 | 1, 4, 5 | sylancl 588 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 3 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 3))) |
7 | simpr 487 | . . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
8 | nn0cn 11901 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
9 | 8cn 11728 | . . . . . . . . . . . 12 ⊢ 8 ∈ ℂ | |
10 | 9 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 8 ∈ ℂ) |
11 | 8, 10 | mulcomd 10656 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑘 · 8) = (8 · 𝑘)) |
12 | 11 | adantl 484 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑘 · 8) = (8 · 𝑘)) |
13 | 12 | oveq1d 7165 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑘 · 8) + 3) = ((8 · 𝑘) + 3)) |
14 | 13 | eqeq2d 2832 | . . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 3) ↔ 𝑃 = ((8 · 𝑘) + 3))) |
15 | 14 | biimpa 479 | . . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 3)) → 𝑃 = ((8 · 𝑘) + 3)) |
16 | 2lgslem2.n | . . . . . . 7 ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) | |
17 | 16 | 2lgslem3b 25967 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝑘) + 3)) → 𝑁 = ((2 · 𝑘) + 1)) |
18 | 7, 15, 17 | syl2an2r 683 | . . . . 5 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 3)) → 𝑁 = ((2 · 𝑘) + 1)) |
19 | oveq1 7157 | . . . . . 6 ⊢ (𝑁 = ((2 · 𝑘) + 1) → (𝑁 mod 2) = (((2 · 𝑘) + 1) mod 2)) | |
20 | nn0z 11999 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
21 | eqidd 2822 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) + 1) = ((2 · 𝑘) + 1)) | |
22 | 2tp1odd 15695 | . . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = ((2 · 𝑘) + 1)) → ¬ 2 ∥ ((2 · 𝑘) + 1)) | |
23 | 20, 21, 22 | syl2anc 586 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((2 · 𝑘) + 1)) |
24 | 2z 12008 | . . . . . . . . . . 11 ⊢ 2 ∈ ℤ | |
25 | 24 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 2 ∈ ℤ) |
26 | 25, 20 | zmulcld 12087 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (2 · 𝑘) ∈ ℤ) |
27 | 26 | peano2zd 12084 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) + 1) ∈ ℤ) |
28 | mod2eq1n2dvds 15690 | . . . . . . . 8 ⊢ (((2 · 𝑘) + 1) ∈ ℤ → ((((2 · 𝑘) + 1) mod 2) = 1 ↔ ¬ 2 ∥ ((2 · 𝑘) + 1))) | |
29 | 27, 28 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((((2 · 𝑘) + 1) mod 2) = 1 ↔ ¬ 2 ∥ ((2 · 𝑘) + 1))) |
30 | 23, 29 | mpbird 259 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (((2 · 𝑘) + 1) mod 2) = 1) |
31 | 19, 30 | sylan9eqr 2878 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑁 = ((2 · 𝑘) + 1)) → (𝑁 mod 2) = 1) |
32 | 7, 18, 31 | syl2an2r 683 | . . . 4 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 3)) → (𝑁 mod 2) = 1) |
33 | 32 | rexlimdva2 3287 | . . 3 ⊢ (𝑃 ∈ ℕ → (∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 3) → (𝑁 mod 2) = 1)) |
34 | 6, 33 | syld 47 | . 2 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 3 → (𝑁 mod 2) = 1)) |
35 | 34 | imp 409 | 1 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 3) → (𝑁 mod 2) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 / cdiv 11291 ℕcn 11632 2c2 11686 3c3 11687 4c4 11688 8c8 11692 ℕ0cn0 11891 ℤcz 11975 ℝ+crp 12383 ⌊cfl 13154 mod cmo 13231 ∥ cdvds 15601 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 ax-pre-sup 10609 |
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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fl 13156 df-mod 13232 df-dvds 15602 |
This theorem is referenced by: 2lgslem3 25974 |
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