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Mirrors > Home > MPE Home > Th. List > 2lgslem3a1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 2lgslem3 25129. (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 11299 | . . . 4 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
2 | 8nn 11191 | . . . . 5 ⊢ 8 ∈ ℕ | |
3 | nnrp 11842 | . . . . 5 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 8 ∈ ℝ+ |
5 | modmuladdnn0 12714 | . . . 4 ⊢ ((𝑃 ∈ ℕ0 ∧ 8 ∈ ℝ+) → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) | |
6 | 1, 4, 5 | sylancl 694 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) |
7 | simpr 477 | . . . . . 6 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
8 | nn0cn 11302 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
9 | 8cn 11106 | . . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ | |
10 | 9 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → 8 ∈ ℂ) |
11 | 8, 10 | mulcomd 10061 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (𝑘 · 8) = (8 · 𝑘)) |
12 | 11 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑘 · 8) = (8 · 𝑘)) |
13 | 12 | oveq1d 6665 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑘 · 8) + 1) = ((8 · 𝑘) + 1)) |
14 | 13 | eqeq2d 2632 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 1) ↔ 𝑃 = ((8 · 𝑘) + 1))) |
15 | 14 | biimpa 501 | . . . . . . 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 25121 | . . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝑘) + 1)) → 𝑁 = (2 · 𝑘)) |
18 | 7, 15, 17 | syl2an2r 876 | . . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑁 = (2 · 𝑘)) |
19 | oveq1 6657 | . . . . . . 7 ⊢ (𝑁 = (2 · 𝑘) → (𝑁 mod 2) = ((2 · 𝑘) mod 2)) | |
20 | 2cnd 11093 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 2 ∈ ℂ) | |
21 | 20, 8 | mulcomd 10061 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (2 · 𝑘) = (𝑘 · 2)) |
22 | 21 | oveq1d 6665 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = ((𝑘 · 2) mod 2)) |
23 | nn0z 11400 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
24 | 2rp 11837 | . . . . . . . . 9 ⊢ 2 ∈ ℝ+ | |
25 | mulmod0 12676 | . . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ 2 ∈ ℝ+) → ((𝑘 · 2) mod 2) = 0) | |
26 | 23, 24, 25 | sylancl 694 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 · 2) mod 2) = 0) |
27 | 22, 26 | eqtrd 2656 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = 0) |
28 | 19, 27 | sylan9eqr 2678 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑁 = (2 · 𝑘)) → (𝑁 mod 2) = 0) |
29 | 7, 18, 28 | syl2an2r 876 | . . . . 5 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → (𝑁 mod 2) = 0) |
30 | 29 | ex 450 | . . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 1) → (𝑁 mod 2) = 0)) |
31 | 30 | rexlimdva 3031 | . . 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 445 | 1 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 4c4 11072 8c8 11076 ℕ0cn0 11292 ℤcz 11377 ℝ+crp 11832 ⌊cfl 12591 mod cmo 12668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fl 12593 df-mod 12669 |
This theorem is referenced by: 2lgslem3 25129 |
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