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Mirrors > Home > MPE Home > Th. List > Mathboxes > lighneallem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for lighneal 45024. (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 12344 | . . . . 5 ⊢ 2 ∈ ℤ | |
2 | simp2 1136 | . . . . 5 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ) | |
3 | iddvdsexp 15979 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 2 ∥ (2↑𝑀)) | |
4 | 1, 2, 3 | sylancr 587 | . . . 4 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 2 ∥ (2↑𝑀)) |
5 | oveq1 7276 | . . . . . 6 ⊢ (𝑃 = 2 → (𝑃↑𝑀) = (2↑𝑀)) | |
6 | 5 | breq2d 5091 | . . . . 5 ⊢ (𝑃 = 2 → (2 ∥ (𝑃↑𝑀) ↔ 2 ∥ (2↑𝑀))) |
7 | 6 | 3ad2ant1 1132 | . . . 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 15979 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 2 ∥ (2↑𝑁)) | |
10 | 1, 9 | mpan 687 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 2 ∥ (2↑𝑁)) |
11 | 10 | notnotd 144 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ ¬ 2 ∥ (2↑𝑁)) |
12 | 2nn 12038 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ) |
14 | nnnn0 12232 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
15 | 13, 14 | nnexpcld 13950 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
16 | 15 | nnzd 12416 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
17 | oddm1even 16042 | . . . . . 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 1134 | . . 3 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ¬ 2 ∥ ((2↑𝑁) − 1)) |
21 | nbrne1 5098 | . . 3 ⊢ ((2 ∥ (𝑃↑𝑀) ∧ ¬ 2 ∥ ((2↑𝑁) − 1)) → (𝑃↑𝑀) ≠ ((2↑𝑁) − 1)) | |
22 | 8, 20, 21 | syl2anc 584 | . 2 ⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ≠ ((2↑𝑁) − 1)) |
23 | 22 | necomd 3001 | 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 1086 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 (class class class)co 7269 1c1 10865 − cmin 11197 ℕcn 11965 2c2 12020 ℤcz 12311 ↑cexp 13772 ∥ cdvds 15953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-div 11625 df-nn 11966 df-2 12028 df-n0 12226 df-z 12312 df-uz 12574 df-seq 13712 df-exp 13773 df-dvds 15954 |
This theorem is referenced by: lighneal 45024 |
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