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Mirrors > Home > ILE Home > Th. List > pw2dvdseulemle | GIF version |
Description: Lemma for pw2dvdseu 11638. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Ref | Expression |
---|---|
pw2dvdseulemle.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
pw2dvdseulemle.a | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
pw2dvdseulemle.b | ⊢ (𝜑 → 𝐵 ∈ ℕ0) |
pw2dvdseulemle.2a | ⊢ (𝜑 → (2↑𝐴) ∥ 𝑁) |
pw2dvdseulemle.n2b | ⊢ (𝜑 → ¬ (2↑(𝐵 + 1)) ∥ 𝑁) |
Ref | Expression |
---|---|
pw2dvdseulemle | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw2dvdseulemle.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | 1 | nn0red 8883 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | pw2dvdseulemle.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ0) | |
4 | 3 | nn0red 8883 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | pw2dvdseulemle.n2b | . . 3 ⊢ (𝜑 → ¬ (2↑(𝐵 + 1)) ∥ 𝑁) | |
6 | 2cnd 8651 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 2 ∈ ℂ) | |
7 | 3 | adantr 272 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℕ0) |
8 | peano2nn0 8869 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ0 → (𝐵 + 1) ∈ ℕ0) | |
9 | 7, 8 | syl 14 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 + 1) ∈ ℕ0) |
10 | 1 | adantr 272 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℕ0) |
11 | simpr 109 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 < 𝐴) | |
12 | nn0ltp1le 8968 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐴 ∈ ℕ0) → (𝐵 < 𝐴 ↔ (𝐵 + 1) ≤ 𝐴)) | |
13 | 7, 10, 12 | syl2anc 406 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 < 𝐴 ↔ (𝐵 + 1) ≤ 𝐴)) |
14 | 11, 13 | mpbid 146 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 + 1) ≤ 𝐴) |
15 | nn0sub2 8976 | . . . . . . 7 ⊢ (((𝐵 + 1) ∈ ℕ0 ∧ 𝐴 ∈ ℕ0 ∧ (𝐵 + 1) ≤ 𝐴) → (𝐴 − (𝐵 + 1)) ∈ ℕ0) | |
16 | 9, 10, 14, 15 | syl3anc 1184 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴 − (𝐵 + 1)) ∈ ℕ0) |
17 | 6, 16, 9 | expaddd 10267 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑((𝐵 + 1) + (𝐴 − (𝐵 + 1)))) = ((2↑(𝐵 + 1)) · (2↑(𝐴 − (𝐵 + 1))))) |
18 | 9 | nn0cnd 8884 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 + 1) ∈ ℂ) |
19 | 10 | nn0cnd 8884 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℂ) |
20 | 18, 19 | pncan3d 7947 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐵 + 1) + (𝐴 − (𝐵 + 1))) = 𝐴) |
21 | 20 | oveq2d 5722 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑((𝐵 + 1) + (𝐴 − (𝐵 + 1)))) = (2↑𝐴)) |
22 | pw2dvdseulemle.2a | . . . . . . 7 ⊢ (𝜑 → (2↑𝐴) ∥ 𝑁) | |
23 | 22 | adantr 272 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑𝐴) ∥ 𝑁) |
24 | 21, 23 | eqbrtrd 3895 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑((𝐵 + 1) + (𝐴 − (𝐵 + 1)))) ∥ 𝑁) |
25 | 17, 24 | eqbrtrrd 3897 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((2↑(𝐵 + 1)) · (2↑(𝐴 − (𝐵 + 1)))) ∥ 𝑁) |
26 | 2nn 8733 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
27 | 26 | a1i 9 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 2 ∈ ℕ) |
28 | 27, 9 | nnexpcld 10287 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐵 + 1)) ∈ ℕ) |
29 | 28 | nnzd 9024 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐵 + 1)) ∈ ℤ) |
30 | 27, 16 | nnexpcld 10287 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐴 − (𝐵 + 1))) ∈ ℕ) |
31 | 30 | nnzd 9024 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐴 − (𝐵 + 1))) ∈ ℤ) |
32 | pw2dvdseulemle.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
33 | 32 | adantr 272 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝑁 ∈ ℕ) |
34 | 33 | nnzd 9024 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝑁 ∈ ℤ) |
35 | muldvds1 11313 | . . . . 5 ⊢ (((2↑(𝐵 + 1)) ∈ ℤ ∧ (2↑(𝐴 − (𝐵 + 1))) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2↑(𝐵 + 1)) · (2↑(𝐴 − (𝐵 + 1)))) ∥ 𝑁 → (2↑(𝐵 + 1)) ∥ 𝑁)) | |
36 | 29, 31, 34, 35 | syl3anc 1184 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (((2↑(𝐵 + 1)) · (2↑(𝐴 − (𝐵 + 1)))) ∥ 𝑁 → (2↑(𝐵 + 1)) ∥ 𝑁)) |
37 | 25, 36 | mpd 13 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐵 + 1)) ∥ 𝑁) |
38 | 5, 37 | mtand 632 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
39 | 2, 4, 38 | nltled 7754 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1448 class class class wbr 3875 (class class class)co 5706 1c1 7501 + caddc 7503 · cmul 7505 < clt 7672 ≤ cle 7673 − cmin 7804 ℕcn 8578 2c2 8629 ℕ0cn0 8829 ℤcz 8906 ↑cexp 10133 ∥ cdvds 11288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-n0 8830 df-z 8907 df-uz 9177 df-seqfrec 10060 df-exp 10134 df-dvds 11289 |
This theorem is referenced by: pw2dvdseu 11638 |
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