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Mirrors > Home > MPE Home > Th. List > 2lgslem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgs 25685. (Contributed by AV, 20-Jun-2021.) |
Ref | Expression |
---|---|
2lgslem2.n | ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) |
Ref | Expression |
---|---|
2lgslem2 | ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lgslem2.n | . 2 ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) | |
2 | simpl 475 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℙ) | |
3 | elsng 4455 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ {2} ↔ 𝑃 = 2)) | |
4 | z2even 15580 | . . . . . . . 8 ⊢ 2 ∥ 2 | |
5 | breq2 4933 | . . . . . . . 8 ⊢ (𝑃 = 2 → (2 ∥ 𝑃 ↔ 2 ∥ 2)) | |
6 | 4, 5 | mpbiri 250 | . . . . . . 7 ⊢ (𝑃 = 2 → 2 ∥ 𝑃) |
7 | 3, 6 | syl6bi 245 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ {2} → 2 ∥ 𝑃)) |
8 | 7 | con3dimp 400 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ¬ 𝑃 ∈ {2}) |
9 | 2, 8 | eldifd 3840 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ (ℙ ∖ {2})) |
10 | oddprm 16003 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
11 | 10 | nnzd 11899 | . . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((𝑃 − 1) / 2) ∈ ℤ) |
13 | prmz 15875 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
14 | 13 | zred 11900 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
15 | 4re 11525 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 4 ∈ ℝ) |
17 | 4ne0 11555 | . . . . . . 7 ⊢ 4 ≠ 0 | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 4 ≠ 0) |
19 | 14, 16, 18 | redivcld 11269 | . . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 / 4) ∈ ℝ) |
20 | 19 | flcld 12983 | . . . 4 ⊢ (𝑃 ∈ ℙ → (⌊‘(𝑃 / 4)) ∈ ℤ) |
21 | 20 | adantr 473 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ∈ ℤ) |
22 | 12, 21 | zsubcld 11905 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ) |
23 | 1, 22 | syl5eqel 2870 | 1 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∖ cdif 3826 {csn 4441 class class class wbr 4929 ‘cfv 6188 (class class class)co 6976 ℝcr 10334 0cc0 10335 1c1 10336 − cmin 10670 / cdiv 11098 2c2 11495 4c4 11497 ℤcz 11793 ⌊cfl 12975 ∥ cdvds 15467 ℙcprime 15871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fl 12977 df-seq 13185 df-exp 13245 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-dvds 15468 df-prm 15872 |
This theorem is referenced by: 2lgs 25685 |
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