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Mirrors > Home > MPE Home > Th. List > Mathboxes > lighneallem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for lighneal 46874. (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 12616 | . . . . 5 ⊢ 2 ∈ ℤ | |
2 | simp2 1135 | . . . . 5 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ) | |
3 | iddvdsexp 16248 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 2 ∥ (2↑𝑀)) | |
4 | 1, 2, 3 | sylancr 586 | . . . 4 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 2 ∥ (2↑𝑀)) |
5 | oveq1 7421 | . . . . . 6 ⊢ (𝑃 = 2 → (𝑃↑𝑀) = (2↑𝑀)) | |
6 | 5 | breq2d 5154 | . . . . 5 ⊢ (𝑃 = 2 → (2 ∥ (𝑃↑𝑀) ↔ 2 ∥ (2↑𝑀))) |
7 | 6 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 ∥ (𝑃↑𝑀) ↔ 2 ∥ (2↑𝑀))) |
8 | 4, 7 | mpbird 257 | . . 3 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 2 ∥ (𝑃↑𝑀)) |
9 | iddvdsexp 16248 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 2 ∥ (2↑𝑁)) | |
10 | 1, 9 | mpan 689 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 2 ∥ (2↑𝑁)) |
11 | 10 | notnotd 144 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ ¬ 2 ∥ (2↑𝑁)) |
12 | 2nn 12307 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ) |
14 | nnnn0 12501 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
15 | 13, 14 | nnexpcld 14231 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
16 | 15 | nnzd 12607 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
17 | oddm1even 16311 | . . . . . 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 324 | . . . 4 ⊢ (𝑁 ∈ ℕ → ¬ 2 ∥ ((2↑𝑁) − 1)) |
20 | 19 | 3ad2ant3 1133 | . . 3 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ¬ 2 ∥ ((2↑𝑁) − 1)) |
21 | nbrne1 5161 | . . 3 ⊢ ((2 ∥ (𝑃↑𝑀) ∧ ¬ 2 ∥ ((2↑𝑁) − 1)) → (𝑃↑𝑀) ≠ ((2↑𝑁) − 1)) | |
22 | 8, 20, 21 | syl2anc 583 | . 2 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ≠ ((2↑𝑁) − 1)) |
23 | 22 | necomd 2991 | 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 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 (class class class)co 7414 1c1 11131 − cmin 11466 ℕcn 12234 2c2 12289 ℤcz 12580 ↑cexp 14050 ∥ cdvds 16222 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-n0 12495 df-z 12581 df-uz 12845 df-seq 13991 df-exp 14051 df-dvds 16223 |
This theorem is referenced by: lighneal 46874 |
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