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Mirrors > Home > ILE Home > Th. List > pw2dvdseulemle | GIF version |
Description: Lemma for pw2dvdseu 12203. 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 9261 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | pw2dvdseulemle.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ0) | |
4 | 3 | nn0red 9261 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | pw2dvdseulemle.n2b | . . 3 ⊢ (𝜑 → ¬ (2↑(𝐵 + 1)) ∥ 𝑁) | |
6 | 2cnd 9023 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 2 ∈ ℂ) | |
7 | 3 | adantr 276 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℕ0) |
8 | peano2nn0 9247 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ0 → (𝐵 + 1) ∈ ℕ0) | |
9 | 7, 8 | syl 14 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 + 1) ∈ ℕ0) |
10 | 1 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℕ0) |
11 | simpr 110 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 < 𝐴) | |
12 | nn0ltp1le 9346 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐴 ∈ ℕ0) → (𝐵 < 𝐴 ↔ (𝐵 + 1) ≤ 𝐴)) | |
13 | 7, 10, 12 | syl2anc 411 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 < 𝐴 ↔ (𝐵 + 1) ≤ 𝐴)) |
14 | 11, 13 | mpbid 147 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 + 1) ≤ 𝐴) |
15 | nn0sub2 9357 | . . . . . . 7 ⊢ (((𝐵 + 1) ∈ ℕ0 ∧ 𝐴 ∈ ℕ0 ∧ (𝐵 + 1) ≤ 𝐴) → (𝐴 − (𝐵 + 1)) ∈ ℕ0) | |
16 | 9, 10, 14, 15 | syl3anc 1249 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴 − (𝐵 + 1)) ∈ ℕ0) |
17 | 6, 16, 9 | expaddd 10690 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑((𝐵 + 1) + (𝐴 − (𝐵 + 1)))) = ((2↑(𝐵 + 1)) · (2↑(𝐴 − (𝐵 + 1))))) |
18 | 9 | nn0cnd 9262 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐵 + 1) ∈ ℂ) |
19 | 10 | nn0cnd 9262 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℂ) |
20 | 18, 19 | pncan3d 8302 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐵 + 1) + (𝐴 − (𝐵 + 1))) = 𝐴) |
21 | 20 | oveq2d 5913 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑((𝐵 + 1) + (𝐴 − (𝐵 + 1)))) = (2↑𝐴)) |
22 | pw2dvdseulemle.2a | . . . . . . 7 ⊢ (𝜑 → (2↑𝐴) ∥ 𝑁) | |
23 | 22 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑𝐴) ∥ 𝑁) |
24 | 21, 23 | eqbrtrd 4040 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑((𝐵 + 1) + (𝐴 − (𝐵 + 1)))) ∥ 𝑁) |
25 | 17, 24 | eqbrtrrd 4042 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((2↑(𝐵 + 1)) · (2↑(𝐴 − (𝐵 + 1)))) ∥ 𝑁) |
26 | 2nn 9111 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
27 | 26 | a1i 9 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 2 ∈ ℕ) |
28 | 27, 9 | nnexpcld 10710 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐵 + 1)) ∈ ℕ) |
29 | 28 | nnzd 9405 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐵 + 1)) ∈ ℤ) |
30 | 27, 16 | nnexpcld 10710 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐴 − (𝐵 + 1))) ∈ ℕ) |
31 | 30 | nnzd 9405 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (2↑(𝐴 − (𝐵 + 1))) ∈ ℤ) |
32 | pw2dvdseulemle.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
33 | 32 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝑁 ∈ ℕ) |
34 | 33 | nnzd 9405 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝑁 ∈ ℤ) |
35 | muldvds1 11858 | . . . . 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 1249 | . . . 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 666 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
39 | 2, 4, 38 | nltled 8109 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5897 1c1 7843 + caddc 7845 · cmul 7847 < clt 8023 ≤ cle 8024 − cmin 8159 ℕcn 8950 2c2 9001 ℕ0cn0 9207 ℤcz 9284 ↑cexp 10553 ∥ cdvds 11829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-n0 9208 df-z 9285 df-uz 9560 df-seqfrec 10479 df-exp 10554 df-dvds 11830 |
This theorem is referenced by: pw2dvdseu 12203 |
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