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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0dig2nn0e | Structured version Visualization version GIF version |
Description: The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.) |
Ref | Expression |
---|---|
0dig2nn0e | ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 11704 | . . . 4 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → 2 ∈ ℕ) |
3 | 0nn0 11906 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → 0 ∈ ℕ0) |
5 | nn0rp0 12837 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0[,)+∞)) | |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → 𝑁 ∈ (0[,)+∞)) |
7 | nn0digval 44654 | . . 3 ⊢ ((2 ∈ ℕ ∧ 0 ∈ ℕ0 ∧ 𝑁 ∈ (0[,)+∞)) → (0(digit‘2)𝑁) = ((⌊‘(𝑁 / (2↑0))) mod 2)) | |
8 | 2, 4, 6, 7 | syl3anc 1367 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = ((⌊‘(𝑁 / (2↑0))) mod 2)) |
9 | 2cn 11706 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
10 | exp0 13427 | . . . . . . . 8 ⊢ (2 ∈ ℂ → (2↑0) = 1) | |
11 | 9, 10 | mp1i 13 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (2↑0) = 1) |
12 | 11 | oveq2d 7166 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (𝑁 / (2↑0)) = (𝑁 / 1)) |
13 | nn0cn 11901 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
14 | 13 | div1d 11402 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 / 1) = 𝑁) |
15 | 14 | adantr 483 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (𝑁 / 1) = 𝑁) |
16 | 12, 15 | eqtrd 2856 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (𝑁 / (2↑0)) = 𝑁) |
17 | 16 | fveq2d 6668 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (⌊‘(𝑁 / (2↑0))) = (⌊‘𝑁)) |
18 | 17 | oveq1d 7165 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → ((⌊‘(𝑁 / (2↑0))) mod 2) = ((⌊‘𝑁) mod 2)) |
19 | nn0z 11999 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
20 | flid 13172 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (⌊‘𝑁) = 𝑁) |
22 | 21 | adantr 483 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (⌊‘𝑁) = 𝑁) |
23 | 22 | oveq1d 7165 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → ((⌊‘𝑁) mod 2) = (𝑁 mod 2)) |
24 | nn0z 11999 | . . . . . 6 ⊢ ((𝑁 / 2) ∈ ℕ0 → (𝑁 / 2) ∈ ℤ) | |
25 | 24 | adantl 484 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (𝑁 / 2) ∈ ℤ) |
26 | nn0re 11900 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
27 | 26 | adantr 483 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → 𝑁 ∈ ℝ) |
28 | 2rp 12388 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
29 | mod0 13238 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ ℝ+) → ((𝑁 mod 2) = 0 ↔ (𝑁 / 2) ∈ ℤ)) | |
30 | 27, 28, 29 | sylancl 588 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → ((𝑁 mod 2) = 0 ↔ (𝑁 / 2) ∈ ℤ)) |
31 | 25, 30 | mpbird 259 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (𝑁 mod 2) = 0) |
32 | 23, 31 | eqtrd 2856 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → ((⌊‘𝑁) mod 2) = 0) |
33 | 18, 32 | eqtrd 2856 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → ((⌊‘(𝑁 / (2↑0))) mod 2) = 0) |
34 | 8, 33 | eqtrd 2856 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 +∞cpnf 10666 / cdiv 11291 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ℤcz 11975 ℝ+crp 12383 [,)cico 12734 ⌊cfl 13154 mod cmo 13231 ↑cexp 13423 digitcdig 44649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-dig 44650 |
This theorem is referenced by: nn0sumshdiglemA 44673 |
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