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Mirrors > Home > MPE Home > Th. List > Mathboxes > lighneallem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for lighneal 47090. (Contributed by AV, 11-Aug-2021.) |
Ref | Expression |
---|---|
lighneallem1 | ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃↑𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12632 | . . . . 5 ⊢ 2 ∈ ℤ | |
2 | simp2 1134 | . . . . 5 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ) | |
3 | iddvdsexp 16265 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 2 ∥ (2↑𝑀)) | |
4 | 1, 2, 3 | sylancr 585 | . . . 4 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 2 ∥ (2↑𝑀)) |
5 | oveq1 7426 | . . . . . 6 ⊢ (𝑃 = 2 → (𝑃↑𝑀) = (2↑𝑀)) | |
6 | 5 | breq2d 5161 | . . . . 5 ⊢ (𝑃 = 2 → (2 ∥ (𝑃↑𝑀) ↔ 2 ∥ (2↑𝑀))) |
7 | 6 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 ∥ (𝑃↑𝑀) ↔ 2 ∥ (2↑𝑀))) |
8 | 4, 7 | mpbird 256 | . . 3 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 2 ∥ (𝑃↑𝑀)) |
9 | iddvdsexp 16265 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 2 ∥ (2↑𝑁)) | |
10 | 1, 9 | mpan 688 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 2 ∥ (2↑𝑁)) |
11 | 10 | notnotd 144 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ ¬ 2 ∥ (2↑𝑁)) |
12 | 2nn 12323 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ) |
14 | nnnn0 12517 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
15 | 13, 14 | nnexpcld 14248 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
16 | 15 | nnzd 12623 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
17 | oddm1even 16328 | . . . . . 6 ⊢ ((2↑𝑁) ∈ ℤ → (¬ 2 ∥ (2↑𝑁) ↔ 2 ∥ ((2↑𝑁) − 1))) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ (2↑𝑁) ↔ 2 ∥ ((2↑𝑁) − 1))) |
19 | 11, 18 | mtbid 323 | . . . 4 ⊢ (𝑁 ∈ ℕ → ¬ 2 ∥ ((2↑𝑁) − 1)) |
20 | 19 | 3ad2ant3 1132 | . . 3 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ¬ 2 ∥ ((2↑𝑁) − 1)) |
21 | nbrne1 5168 | . . 3 ⊢ ((2 ∥ (𝑃↑𝑀) ∧ ¬ 2 ∥ ((2↑𝑁) − 1)) → (𝑃↑𝑀) ≠ ((2↑𝑁) − 1)) | |
22 | 8, 20, 21 | syl2anc 582 | . 2 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ≠ ((2↑𝑁) − 1)) |
23 | 22 | necomd 2985 | 1 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃↑𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 (class class class)co 7419 1c1 11146 − cmin 11481 ℕcn 12250 2c2 12305 ℤcz 12596 ↑cexp 14067 ∥ cdvds 16239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-seq 14008 df-exp 14068 df-dvds 16240 |
This theorem is referenced by: lighneal 47090 |
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