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Mirrors > Home > MPE Home > Th. List > 2lgslem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgs 26242. (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 486 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℙ) | |
3 | elsng 4541 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ {2} ↔ 𝑃 = 2)) | |
4 | z2even 15894 | . . . . . . . 8 ⊢ 2 ∥ 2 | |
5 | breq2 5043 | . . . . . . . 8 ⊢ (𝑃 = 2 → (2 ∥ 𝑃 ↔ 2 ∥ 2)) | |
6 | 4, 5 | mpbiri 261 | . . . . . . 7 ⊢ (𝑃 = 2 → 2 ∥ 𝑃) |
7 | 3, 6 | syl6bi 256 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ {2} → 2 ∥ 𝑃)) |
8 | 7 | con3dimp 412 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ¬ 𝑃 ∈ {2}) |
9 | 2, 8 | eldifd 3864 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ (ℙ ∖ {2})) |
10 | oddprm 16326 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
11 | 10 | nnzd 12246 | . . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((𝑃 − 1) / 2) ∈ ℤ) |
13 | prmz 16195 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
14 | 13 | zred 12247 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
15 | 4re 11879 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 4 ∈ ℝ) |
17 | 4ne0 11903 | . . . . . . 7 ⊢ 4 ≠ 0 | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 4 ≠ 0) |
19 | 14, 16, 18 | redivcld 11625 | . . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 / 4) ∈ ℝ) |
20 | 19 | flcld 13338 | . . . 4 ⊢ (𝑃 ∈ ℙ → (⌊‘(𝑃 / 4)) ∈ ℤ) |
21 | 20 | adantr 484 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ∈ ℤ) |
22 | 12, 21 | zsubcld 12252 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ) |
23 | 1, 22 | eqeltrid 2835 | 1 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 {csn 4527 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 − cmin 11027 / cdiv 11454 2c2 11850 4c4 11852 ℤcz 12141 ⌊cfl 13330 ∥ cdvds 15778 ℙcprime 16191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-fl 13332 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-dvds 15779 df-prm 16192 |
This theorem is referenced by: 2lgs 26242 |
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