![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2lgslem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgs 27433. (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 481 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℙ) | |
3 | elsng 4637 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ {2} ↔ 𝑃 = 2)) | |
4 | z2even 16367 | . . . . . . . 8 ⊢ 2 ∥ 2 | |
5 | breq2 5149 | . . . . . . . 8 ⊢ (𝑃 = 2 → (2 ∥ 𝑃 ↔ 2 ∥ 2)) | |
6 | 4, 5 | mpbiri 257 | . . . . . . 7 ⊢ (𝑃 = 2 → 2 ∥ 𝑃) |
7 | 3, 6 | biimtrdi 252 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ {2} → 2 ∥ 𝑃)) |
8 | 7 | con3dimp 407 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ¬ 𝑃 ∈ {2}) |
9 | 2, 8 | eldifd 3957 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ (ℙ ∖ {2})) |
10 | oddprm 16807 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ) | |
11 | 10 | nnzd 12631 | . . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((𝑃 − 1) / 2) ∈ ℤ) |
13 | prmz 16671 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
14 | 13 | zred 12712 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
15 | 4re 12342 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 4 ∈ ℝ) |
17 | 4ne0 12366 | . . . . . . 7 ⊢ 4 ≠ 0 | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 4 ≠ 0) |
19 | 14, 16, 18 | redivcld 12087 | . . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 / 4) ∈ ℝ) |
20 | 19 | flcld 13812 | . . . 4 ⊢ (𝑃 ∈ ℙ → (⌊‘(𝑃 / 4)) ∈ ℤ) |
21 | 20 | adantr 479 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ∈ ℤ) |
22 | 12, 21 | zsubcld 12717 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ) |
23 | 1, 22 | eqeltrid 2830 | 1 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3943 {csn 4623 class class class wbr 5145 ‘cfv 6546 (class class class)co 7416 ℝcr 11148 0cc0 11149 1c1 11150 − cmin 11485 / cdiv 11912 2c2 12313 4c4 12315 ℤcz 12604 ⌊cfl 13804 ∥ cdvds 16251 ℙcprime 16667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-n0 12519 df-z 12605 df-uz 12869 df-rp 13023 df-fl 13806 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-prm 16668 |
This theorem is referenced by: 2lgs 27433 |
Copyright terms: Public domain | W3C validator |