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Mirrors > Home > MPE Home > Th. List > 2lgslem3a1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 2lgslem3 27463. (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 12531 | . . . 4 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
2 | 8nn 12359 | . . . . 5 ⊢ 8 ∈ ℕ | |
3 | nnrp 13044 | . . . . 5 ⊢ (8 ∈ ℕ → 8 ∈ ℝ+) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 8 ∈ ℝ+ |
5 | modmuladdnn0 13953 | . . . 4 ⊢ ((𝑃 ∈ ℕ0 ∧ 8 ∈ ℝ+) → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) | |
6 | 1, 4, 5 | sylancl 586 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → ∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1))) |
7 | simpr 484 | . . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
8 | nn0cn 12534 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
9 | 8cn 12361 | . . . . . . . . . . . 12 ⊢ 8 ∈ ℂ | |
10 | 9 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 8 ∈ ℂ) |
11 | 8, 10 | mulcomd 11280 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑘 · 8) = (8 · 𝑘)) |
12 | 11 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑘 · 8) = (8 · 𝑘)) |
13 | 12 | oveq1d 7446 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑘 · 8) + 1) = ((8 · 𝑘) + 1)) |
14 | 13 | eqeq2d 2746 | . . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃 = ((𝑘 · 8) + 1) ↔ 𝑃 = ((8 · 𝑘) + 1))) |
15 | 14 | biimpa 476 | . . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑃 = ((8 · 𝑘) + 1)) |
16 | 2lgslem2.n | . . . . . . 7 ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) | |
17 | 16 | 2lgslem3a 27455 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝑘) + 1)) → 𝑁 = (2 · 𝑘)) |
18 | 7, 15, 17 | syl2an2r 685 | . . . . 5 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → 𝑁 = (2 · 𝑘)) |
19 | oveq1 7438 | . . . . . 6 ⊢ (𝑁 = (2 · 𝑘) → (𝑁 mod 2) = ((2 · 𝑘) mod 2)) | |
20 | 2cnd 12342 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 2 ∈ ℂ) | |
21 | 20, 8 | mulcomd 11280 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2 · 𝑘) = (𝑘 · 2)) |
22 | 21 | oveq1d 7446 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = ((𝑘 · 2) mod 2)) |
23 | nn0z 12636 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
24 | 2rp 13037 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
25 | mulmod0 13914 | . . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ 2 ∈ ℝ+) → ((𝑘 · 2) mod 2) = 0) | |
26 | 23, 24, 25 | sylancl 586 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 · 2) mod 2) = 0) |
27 | 22, 26 | eqtrd 2775 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 → ((2 · 𝑘) mod 2) = 0) |
28 | 19, 27 | sylan9eqr 2797 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑁 = (2 · 𝑘)) → (𝑁 mod 2) = 0) |
29 | 7, 18, 28 | syl2an2r 685 | . . . 4 ⊢ (((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) ∧ 𝑃 = ((𝑘 · 8) + 1)) → (𝑁 mod 2) = 0) |
30 | 29 | rexlimdva2 3155 | . . 3 ⊢ (𝑃 ∈ ℕ → (∃𝑘 ∈ ℕ0 𝑃 = ((𝑘 · 8) + 1) → (𝑁 mod 2) = 0)) |
31 | 6, 30 | syld 47 | . 2 ⊢ (𝑃 ∈ ℕ → ((𝑃 mod 8) = 1 → (𝑁 mod 2) = 0)) |
32 | 31 | imp 406 | 1 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 4c4 12321 8c8 12325 ℕ0cn0 12524 ℤcz 12611 ℝ+crp 13032 ⌊cfl 13827 mod cmo 13906 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-fl 13829 df-mod 13907 |
This theorem is referenced by: 2lgslem3 27463 |
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