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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 12499 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
2 | 1 | nn0zi 12594 | . . . . . . . 8 ⊢ 5 ∈ ℤ |
3 | 2z 12601 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
4 | dvdsmul2 16229 | . . . . . . . 8 ⊢ ((5 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (5 · 2)) | |
5 | 2, 3, 4 | mp2an 689 | . . . . . . 7 ⊢ 2 ∥ (5 · 2) |
6 | 5t2e10 12784 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
7 | 5, 6 | breqtri 5173 | . . . . . 6 ⊢ 2 ∥ ;10 |
8 | 10nn0 12702 | . . . . . . . 8 ⊢ ;10 ∈ ℕ0 | |
9 | 8 | nn0zi 12594 | . . . . . . 7 ⊢ ;10 ∈ ℤ |
10 | dec2dvds.1 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
11 | 10 | nn0zi 12594 | . . . . . . 7 ⊢ 𝐴 ∈ ℤ |
12 | dvdsmultr1 16246 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ ;10 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 ∥ ;10 → 2 ∥ (;10 · 𝐴))) | |
13 | 3, 9, 11, 12 | mp3an 1460 | . . . . . 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 12594 | . . . . . . 7 ⊢ 𝐵 ∈ ℤ |
17 | dvdsmul2 16229 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (𝐵 · 2)) | |
18 | 16, 3, 17 | mp2an 689 | . . . . . 6 ⊢ 2 ∥ (𝐵 · 2) |
19 | dec2dvds.3 | . . . . . 6 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | breqtri 5173 | . . . . 5 ⊢ 2 ∥ 𝐶 |
21 | 8, 10 | nn0mulcli 12517 | . . . . . . 7 ⊢ (;10 · 𝐴) ∈ ℕ0 |
22 | 21 | nn0zi 12594 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℤ |
23 | 2nn0 12496 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
24 | 15, 23 | nn0mulcli 12517 | . . . . . . . 8 ⊢ (𝐵 · 2) ∈ ℕ0 |
25 | 19, 24 | eqeltrri 2829 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 |
26 | 25 | nn0zi 12594 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
27 | dvds2add 16240 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ (;10 · 𝐴) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶))) | |
28 | 3, 22, 26, 27 | mp3an 1460 | . . . . 5 ⊢ ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶)) |
29 | 14, 20, 28 | mp2an 689 | . . . 4 ⊢ 2 ∥ ((;10 · 𝐴) + 𝐶) |
30 | dfdec10 12687 | . . . 4 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
31 | 29, 30 | breqtrri 5175 | . . 3 ⊢ 2 ∥ ;𝐴𝐶 |
32 | 10, 25 | deccl 12699 | . . . . 5 ⊢ ;𝐴𝐶 ∈ ℕ0 |
33 | 32 | nn0zi 12594 | . . . 4 ⊢ ;𝐴𝐶 ∈ ℤ |
34 | 2nn 12292 | . . . 4 ⊢ 2 ∈ ℕ | |
35 | 1lt2 12390 | . . . 4 ⊢ 1 < 2 | |
36 | ndvdsp1 16361 | . . . 4 ⊢ ((;𝐴𝐶 ∈ ℤ ∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1))) | |
37 | 33, 34, 35, 36 | mp3an 1460 | . . 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 2740 | . . . 4 ⊢ (𝐶 + 1) = 𝐷 |
41 | eqid 2731 | . . . 4 ⊢ ;𝐴𝐶 = ;𝐴𝐶 | |
42 | 10, 25, 40, 41 | decsuc 12715 | . . 3 ⊢ (;𝐴𝐶 + 1) = ;𝐴𝐷 |
43 | 42 | breq2i 5156 | . 2 ⊢ (2 ∥ (;𝐴𝐶 + 1) ↔ 2 ∥ ;𝐴𝐷) |
44 | 38, 43 | mtbi 322 | 1 ⊢ ¬ 2 ∥ ;𝐴𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 < clt 11255 ℕcn 12219 2c2 12274 5c5 12277 ℕ0cn0 12479 ℤcz 12565 ;cdc 12684 ∥ cdvds 16204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 |
This theorem is referenced by: 11prm 17055 13prm 17056 17prm 17057 19prm 17058 23prm 17059 37prm 17061 43prm 17062 83prm 17063 139prm 17064 163prm 17065 317prm 17066 631prm 17067 257prm 46540 139prmALT 46575 31prm 46576 127prm 46578 |
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