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Mirrors > Home > MPE Home > Th. List > dec2dvds | Structured version Visualization version GIF version |
Description: Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
dec2dvds.1 | ⊢ 𝐴 ∈ ℕ0 |
dec2dvds.2 | ⊢ 𝐵 ∈ ℕ0 |
dec2dvds.3 | ⊢ (𝐵 · 2) = 𝐶 |
dec2dvds.4 | ⊢ 𝐷 = (𝐶 + 1) |
Ref | Expression |
---|---|
dec2dvds | ⊢ ¬ 2 ∥ ;𝐴𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 11905 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
2 | 1 | nn0zi 11995 | . . . . . . . 8 ⊢ 5 ∈ ℤ |
3 | 2z 12002 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
4 | dvdsmul2 15620 | . . . . . . . 8 ⊢ ((5 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (5 · 2)) | |
5 | 2, 3, 4 | mp2an 688 | . . . . . . 7 ⊢ 2 ∥ (5 · 2) |
6 | 5t2e10 12186 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
7 | 5, 6 | breqtri 5082 | . . . . . 6 ⊢ 2 ∥ ;10 |
8 | 10nn0 12104 | . . . . . . . 8 ⊢ ;10 ∈ ℕ0 | |
9 | 8 | nn0zi 11995 | . . . . . . 7 ⊢ ;10 ∈ ℤ |
10 | dec2dvds.1 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
11 | 10 | nn0zi 11995 | . . . . . . 7 ⊢ 𝐴 ∈ ℤ |
12 | dvdsmultr1 15635 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ ;10 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 ∥ ;10 → 2 ∥ (;10 · 𝐴))) | |
13 | 3, 9, 11, 12 | mp3an 1452 | . . . . . 6 ⊢ (2 ∥ ;10 → 2 ∥ (;10 · 𝐴)) |
14 | 7, 13 | ax-mp 5 | . . . . 5 ⊢ 2 ∥ (;10 · 𝐴) |
15 | dec2dvds.2 | . . . . . . . 8 ⊢ 𝐵 ∈ ℕ0 | |
16 | 15 | nn0zi 11995 | . . . . . . 7 ⊢ 𝐵 ∈ ℤ |
17 | dvdsmul2 15620 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (𝐵 · 2)) | |
18 | 16, 3, 17 | mp2an 688 | . . . . . 6 ⊢ 2 ∥ (𝐵 · 2) |
19 | dec2dvds.3 | . . . . . 6 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | breqtri 5082 | . . . . 5 ⊢ 2 ∥ 𝐶 |
21 | 8, 10 | nn0mulcli 11923 | . . . . . . 7 ⊢ (;10 · 𝐴) ∈ ℕ0 |
22 | 21 | nn0zi 11995 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℤ |
23 | 2nn0 11902 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
24 | 15, 23 | nn0mulcli 11923 | . . . . . . . 8 ⊢ (𝐵 · 2) ∈ ℕ0 |
25 | 19, 24 | eqeltrri 2907 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 |
26 | 25 | nn0zi 11995 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
27 | dvds2add 15631 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ (;10 · 𝐴) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶))) | |
28 | 3, 22, 26, 27 | mp3an 1452 | . . . . 5 ⊢ ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶)) |
29 | 14, 20, 28 | mp2an 688 | . . . 4 ⊢ 2 ∥ ((;10 · 𝐴) + 𝐶) |
30 | dfdec10 12089 | . . . 4 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
31 | 29, 30 | breqtrri 5084 | . . 3 ⊢ 2 ∥ ;𝐴𝐶 |
32 | 10, 25 | deccl 12101 | . . . . 5 ⊢ ;𝐴𝐶 ∈ ℕ0 |
33 | 32 | nn0zi 11995 | . . . 4 ⊢ ;𝐴𝐶 ∈ ℤ |
34 | 2nn 11698 | . . . 4 ⊢ 2 ∈ ℕ | |
35 | 1lt2 11796 | . . . 4 ⊢ 1 < 2 | |
36 | ndvdsp1 15750 | . . . 4 ⊢ ((;𝐴𝐶 ∈ ℤ ∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1))) | |
37 | 33, 34, 35, 36 | mp3an 1452 | . . 3 ⊢ (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1)) |
38 | 31, 37 | ax-mp 5 | . 2 ⊢ ¬ 2 ∥ (;𝐴𝐶 + 1) |
39 | dec2dvds.4 | . . . . 5 ⊢ 𝐷 = (𝐶 + 1) | |
40 | 39 | eqcomi 2827 | . . . 4 ⊢ (𝐶 + 1) = 𝐷 |
41 | eqid 2818 | . . . 4 ⊢ ;𝐴𝐶 = ;𝐴𝐶 | |
42 | 10, 25, 40, 41 | decsuc 12117 | . . 3 ⊢ (;𝐴𝐶 + 1) = ;𝐴𝐷 |
43 | 42 | breq2i 5065 | . 2 ⊢ (2 ∥ (;𝐴𝐶 + 1) ↔ 2 ∥ ;𝐴𝐷) |
44 | 38, 43 | mtbi 323 | 1 ⊢ ¬ 2 ∥ ;𝐴𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 ℕcn 11626 2c2 11680 5c5 11683 ℕ0cn0 11885 ℤcz 11969 ;cdc 12086 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 |
This theorem is referenced by: 11prm 16436 13prm 16437 17prm 16438 19prm 16439 23prm 16440 37prm 16442 43prm 16443 83prm 16444 139prm 16445 163prm 16446 317prm 16447 631prm 16448 257prm 43600 139prmALT 43636 31prm 43637 127prm 43640 |
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